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Homotopy Coherence and Higher Semiadditivity

Mon, 09 Mar 2026 00:00:00 GMT

import * as images from "../../../images/liorcoherence" ### Goal A setting where we can do homotopy theory with monoids within a category. - Topological monoids (common approach to $\infty$-category of objects) are too rigid - Monoids in the homotopy category of $\mathsf{Top}$ are too loose. A middle ground is the category of **homotopy coherent** monoids, tensor categories with commuting Stasheff pentagons and higher diagrams:

### Framing associativity as a coherence problem For a homotopy theorist, associativity of multiplication for a monoid $M$ is connected to existence of extensions fitting in a diagram

(using the appropriate sphere/disk objects in the category, eg coming from initial/terminal objects and categorical constructions). More generally, we have a family of extension problems

and we define a **homotopy coherent monoid** to be a monoid equipped with solutions to these extension problems for all $n$. **Inverses**: Let $M \times M \xrightarrow{s} M \times M$ be the *shear* map $(x,y) \mapsto (x, xy)$. Then $M$ is a group if and only if $s$ is a bijection. **Definition**: A **homotopy coherent (hc) group** is a hc monoid such that $s$ is a homotopy equivalence. Example: For $Y$ a topological space, $X = \Omega Y = \text{Map}_{*}(\mathbb{S}^1, Y)$ (the basepoint-preserving maps for fixed basepoint of $Y$) is a hc group. (rough statement of) **Theorem:** Homotopy coherent groups correspond bijectively to loop spaces. Recall that the Eckmann-Hilton argument says that if a space $X = X_0$ is isomorphic to $\Omega^2 X_2$ for some other space $X_2$, $\pi_{2}(X) = \pi_{0}(X_{2})$ is abelian. Since fundamental groups may be interpreted as maps from $\mathbb{S}^n$ into $X$, we see that if there are spaces $X_m$ such that $X = \Omega^m X_m$, the homotopy groups $\pi_m(X)$ will be abelian for $m \geq 2$. Combining with the above theorem, this should imply that homotopy coherent commutative groups correspond to infinite loopspaces. Examples: - Let $(\mathscr{C}, \otimes, \mathbb{1})$ be a symmetric monoidal category. Then we usually assume $$ X\otimes (Y \otimes Z) \simeq (X \otimes Y) \otimes Z $$ up to isomorphism, together with higher coherence of the isomorphisms (Stasheff pentagon is used as an associator and implies higher diagrams). These coherences can be thought as belonging to the subcategory consisting of objects and invertible morphisms (so there is a groupoid perspective on this). - Recall that we have functors

taking the fundamental groupoid in one direction and the geometric realisation in the other. For example, the geometric realisation of the *classifying groupoid* $\mathbb{B}G$ (category with one object and $G$ as its group of automorphisms) is the **classifying space** $|\mathbb{B}G| = BG$. Applying the geometric realisation construction to the category $\mathsf{Fin}^{\simeq}$ of finite sets up to isomorphism (objects: one for each natural number $n$, no morphisms between different sets and $\text{Hom}(n,n) = \Sigma_{n}$ the symmetric group on $n$ elements) gives $$ |\mathsf{Fin}^{\simeq}| = \coprod_{n \in \mathbb{N}} B\Sigma_{n} =: \mathbb{M}, $$ which has a homotopy coherent commutative monoid structure given by disjoint union. The resulting monoid is the free monoid on one element (note: the monoid itself is not commutative but its higher associations are). Applying $\pi_{0}$ gives the monoid $(\mathbb{N}, +)$, which we can group-complete to get $(\mathbb{Z}, +)$. The corresponding 'group completion' object of $\mathbb{M}$ turns out to be the *sphere spectrum* $$ \mathbb{S} = \varinjlim (\mathbb{S}^0 \to \Omega \mathbb{S}^1 \to \Omega^2 \mathbb{S}^2 \to \dots), $$ in the sense that we have a diagram

Of course, for this to be the right choice of completion we need more than just the same underlying monoid as the group completion, but I didn't catch this detail in the talk. For one hint in this direction, recall that by the Freudenthal suspension theorem, the sequence $$ \pi_{n}(\mathbb{S}) = \varinjlim_{k} \pi_{n}(\Omega^k(\mathbb{S}^k)) = \varinjlim_{k} \pi_{n+k}(\mathbb{S}^k) $$ stabilises, so that if we draw a similar diagram at $\pi_{n}$ we are getting the *stable stems* of the sphere spectrum along the bottom. ## Decategorification One use for homotopy coherent constructions is *decategorification*, reducing the full collection of hc data to structure which is easier to carry around. For a symmetric monoidal category $(\mathscr{C}, \otimes, \mathbb{1})$, the category $\pi_{0}(\mathscr{C})$ of isomorphism classes of objects in $\mathscr{C}$ is a commutative semiring under operations $$ \begin{aligned} \,[X] + [Y] &= [X\cup Y] \\ [X][Y] &= [X \otimes Y], \end{aligned} $$ with group completion $K_{0}(\mathscr{C})$ a commutative ring. We have a **symmetric power** operation $$ S^n[X] = [X^{\otimes n}/\Sigma_{n}] \ (\text{coinvariants}) $$ using the action of $\Sigma_{n}$ on $X^{\otimes n}$, and often we can also form the **alternating power** operation $\bigwedge\nolimits^{n}[X] = [X^{\otimes n}/ \text{sgn}(\Sigma_{n})]$ (when we have a sign representation). If $\mathscr{C}$ is $\mathbb{C}$-linear, we also get a $\lambda$-ring structure on $K_{0}(\mathscr{C})$. Examples - Let $\mathscr{C} = \mathsf{Vect}_{\mathbb{C}}^{BG}$ be the category of complex vector bundles with structure group $G$ over a point. Then $K_0(\mathscr{C}) = \mathsf{Rep}(G)$ inherits a $\lambda$-ring structure, from the alternating power operation. - Let $\mathscr{C} = \mathsf{Fin}^{BG}_{\mathbb{C}}$ be the category of all finite sets admitting $G$-actions. Then $K_{0}(\mathscr{C}) = \mathsf{Burn}(G)$ is a ring, called the **Burnside ring** of $G$, with addition given by disjoint union of sets and multiplication by Cartesian product. Remark: The group completion of $\left( \coprod_{n \in \mathbb{N}} B\mathrm{GL}_{n}(\mathbb{C}), \oplus \right)$ in the category of spectra is an object usually denoted by $ku$, the spectrum of **topological complex K-theory**. The name comes from the fact that for a compact object $X$ in the category, $K_{0}(X) = [X, ku],$ the maps from $X$ to $ku$. With the tensor product $\otimes$ of complex vector space, $ku$ comes equipped with hc commutative ring structure. Example - Power Elements Let $R$ be any commutative ring spectrum (in the topological sense, so we have an infinite suspension diagram of rings). Then an element $x \in R$ can be thought of as an arrow in a diagram of hc objects

Since $\mathbb{M} = \coprod_{n \in \mathbb{N}} B\Sigma_{n}$ we have maps from the point to each one-point space $B\Sigma_{n}$. Now a map $B\Sigma_{n} \to R$ picks out an element $x^n \in R^{B\Sigma_{n}}$ the hc ring of all objects with automorphism group $\Sigma_{n}$ (identified with $\mathsf{Map}(B\Sigma_{n}, R)$), and we have $\pi_{0}(R^{B\Sigma_{n}}) = R^0(B\Sigma_{n}),$ where $R^0 = \pi_{0}(R)$ (so the RHS is the ordinary points of the ordinary ring $R$ which have $\Sigma_{n}$-automorphisms). Coherence is equivalent to the statement that the diagram

commutes,

where the object on the bottom is the quotient groupoid (every object of $R^n$ has $\Sigma_{n}$-automorphisms). Now since we have an arrow $\alpha: R^0(\mathbb{M}) \to R^0$ as above, we get **power elements** $\alpha(x^m) \in R$ for all $m$ from our original diagram. (This is probably only partially correct, I am missing definitions and details to make this precise) I think we want this to explain the power elements/formal group laws of Lubin, Lurie et al - There is a *Thom spectrum* $MO$ such that $\pi_{n}(MO)$ is the category of stably framed smooth $n$-manifolds up to cobordism, and $[X, MO]$ gives *cobordism cohomology* for a topological space $X$. - The *Morava E-theories* form a sequence of commutative ring spectra, with - $E_0 \approx \mathbb{C}$ - $E_1 \approx k\hat{u_{p}}$ and higher $E$'s being related to things like Witt vector cohomology. I think these are used in topology to study torsion phenomena. Example: The Hopf fibration can be constructed by decategorification. In the sphere spectrum, taking $x \in \Omega^2 \mathbb{S}^2$ we have arrows $$ 1 \sim x\cdot x^{-1} \overset{ \ \sigma_{x,x^{-1}}}\sim x^{-1}\cdot x \sim 1 $$ (using a $\Sigma_{2}$-action on the product $x\cdot x^{-1}$); the loop corresponds to the Hopf map $\mathbb{S}^1 \to \Omega^2 \mathbb{S}^2$, ie $\mathbb{S}^3 \xrightarrow{\eta} \mathbb{S}^2$. Orbits: If we have a categorical group action $BG \xrightarrow{Z} \mathscr{C}$ factoring through the one-point space $BG \to \text{pt} \to \mathscr{C}$, then we have $\varinjlim Z = Z//G$, identifying $Z$ with its one-object image. In particular, the limit over $[n] \xrightarrow{\{X_{i}\}_{i \in [n]}} \mathscr{C}$ is given by $\varinjlim_{[n]}X_{i} = X_{1} \cup \dots \cup X_{n}$. ### Semiadditivity **Definition**: A space $X$ is **n-finite** if $\pi_{k}(X, x) = 0$ for all $k > n$ and if all homotopy groups $\pi_k(X, x)$ are finite, for every $x \in X$ and $k \in \mathbb{N}.$ - 0-finite spaces are the finite discrete spaces, up to homotopy equivalence. - 1-finite spaces are unions of classifying spaces $\mathbb{B}G_{i}$, where each $G_i$ is a finite group. **Definition**: An **n-commutative monoid** is a (Segal) space $X$ such that: - For every $n$-finite space $\mathcal{A}$, the left Kan extension $\int_{\mathcal{A}}: X^\mathcal{A} \to X$ exists - There is homotopy-coherent associativity and a homotopy-coherent unit - Analogous to taking the orbits of the induced representation, we have a commuting diagram

Example: If $X = \mathbb{Q}$ (more generally, any divisible commutative monoid) and we have a constant map $BG \xrightarrow{f} \mathbb{Q}$, the left Kan extension is $\int_{BG}f = \frac{1}{|G|}f(x)$. So we see that the extension is providing a generalisation of 'averaging over $G$'. **Definition**: A category is **(-1)-semiadditive** or **pointed** if its intial and terminal object are isomorphic. **Definition**: A category is **(0-)semiadditive** if for all finite collections $X_1, \dots, X_{n}$ of objects, $$ X_{1} \cup \dots \cup X_{n} \cong X_{1} \times \dots \times X_{n} $$ via the canonical morphism $X_{1} \cup \dots \cup X_{n} \to X_{1} \times \dots \times X_{n}$. **Definition** (Hopkins-Lurie): An $\infty$-category $\mathscr{C}$ is **n-semiadditive** if for every diagram $X: A \to \mathscr{C}$ with $A$ an n-finite space, there is a canonical isomorphism $$ \varinjlim_{A} \ X \cong \varprojlim_{A} \ X. $$ If $X, Y \in \mathscr{C}$ are $0$-semiadditive, $f, \, g : X \to Y$ we can construct the diagram

recovering and generalising addition to the homotopy-coherent setting\! **Theorem**: If $\mathscr{C}$ is $n$-semiadditive for all $X, Y \in \mathscr{C}$, $\text{Map}_{\mathscr{C}}(X,Y)$ has a canonical $n$-commutative monoid structure. - In the category $\mathsf{Sp}_{(p)}$ of $p$-localised spectra, there is a sequence of ringed spectra $K(n)$ for $0 \leq n \leq \infty$, called the **Morava K-theories**. We have $K(0) = H(\mathbb{Q})$ the spectrum of rational cohomology, and $K(\infty) = H\mathbb{F}_{p}$ the spectrum of mod $p$ cohomology. - If one knows $K(n)$-locality and support theorems, then one can show that $\mathsf{Sp}_{K(n)} \xhookrightarrow{} \mathsf{Sp}_{(p)}$. **Theorem**(Hopkins-Lurie): The categories $\mathsf{Sp}_{K(n)}$ are $\infty$-semiadditive, for $n < \infty$. **Theorem**(Y.): The categories $\mathsf{Sp}_{T(n)}$ for $n < \infty $ are also $\infty$-semiadditive. - If $R$ is a $K(n)$-(or $T(n)$-)local commutative ring spectrum, all maps $R^{B\Sigma_{n}} \to R$ are of the form $x\mapsto \int_{B\Sigma_{n}}ax^n$, for $a \in R^{B\Sigma_{n}}$. Example: In height $0$ and over $\mathbb{Q}$, all possible power operations are given by $$ a_{0} + a_{1}x + \frac{a_{2}}{2!}x^2 + \dots + \frac{a_{n}}{n!}x^n. $$

Rational Maps

Tue, 03 Mar 2026 00:00:00 GMT

In the category of varieties (and more generally, schemes), morphisms between varieties are determined (at least locally) by $k$-algebra homomorphisms between the rings of regular functions. Another invariant of a variety $X$ is its function field $K(X)$, whose elements are the germs $\langle U, \varphi_{U} \rangle$ with $\varphi_U$ a regular function on the open set $U$. Multiplying germs is then achieved sheaf-theoretically, $$ \langle U, \varphi_{U} \rangle\ \cdot \langle V, \varphi_{V} \rangle = \langle U\cap V, \varphi_{U} \cdot \varphi_{V} \rangle. $$ Rational maps answer two questions for us:

Is there a corresponding geometric category for the maps between function fields?
Locally, regular functions are quotients of polynomials. To what extent can we obtain a similar global description of the regular functions?

**Definition**: Let $X, Y$ be varieties. A **rational map** $X \to Y$ is an equivalence class $\langle U, \varphi_{U} \rangle$ with $U \subseteq X$ open and $\varphi_{U} : U \to \varphi_{U}(U) \subseteq Y$ a morphism, where the equivalence relation is that $$ \langle U, \varphi_{U} \rangle \sim \langle V, \varphi_{V} \rangle \iff \varphi_{U}|_{U\cap V} = \varphi_{V}|_{U \cap V}, $$ where $U\cap V \neq 0$. A rational map is **dominant** if it has a class representative $\langle U, \varphi_{U} \rangle$ with dense image in $Y$. **Definition**: A **birational map** is a rational map with a two-sided rational inverse. If there is a birational map between $X$ and $Y$ varieties, we say that $X$ is **birationally equivalent** to $Y$. **Lemma**: Let $\varphi, \ \psi: X \to Y$ be two morphisms agreeing on a nonempty open set $U$. Then $\varphi = \psi$ on all of $X$. **Proof**: Any variety is irreducible, so in particular $U$ in dense in $X$. Let $f: Y \to k$ be regular and consider the closed set $V(f\varphi - f\psi)$ in $X$. We have $U \subseteq V(f\varphi - f\psi)$, so it follows that $V(f\varphi - f\psi) = X$ by density. Now points are closed for any variety, so given $y \in \varphi(X)$ we can choose a regular function $g_{y}: Y \to k$ whose only zero is $y$. For any $x \in \varphi^{-1}(y)$ we have $x \in V(g_{y}\varphi - g_{y}\psi)$ by the above statement, so that $\varphi(x) = \psi(x)$. Letting $y$ vary gives the equality of of functions. $\square$

First phrasing of proof: If $\varphi(x) \neq \psi(x)$, then for $f: Y \to k$ regular we have $f\varphi - f\psi \neq 0$ (as $U$ is a collection of points on which it is nonvanishing) and $V(f\varphi - f\psi)$ is closed in $X$. The complement is an open set containing $U$, contradicting density.

**Note**: This result is telling us a property of the topology of varieties: the diagonal subset $\Delta_{Y} = \{(y,y) \in Y \times Y\}$ of their product is always closed. This condition is called being **separated** in algebraic geometry, and is used in analogy to Hausdorffness. Stated this way, it looks identical to the usual Hausdorff condition, but the topology on the product of varieties is not the product topology. We need to know a little more about the category of varieties to obtain a good answer to (1). **Lemma**: Let $Y \subseteq \mathbb{A}^n$ be the hypersurface $V(f)$, where $f \in k[x_1 \dots, x_n]$. Then the complement $H = \mathbb{A}^n \setminus Y$ is isomorphic to $V(x_{n+1}f-1) \subseteq \mathbb{A}^{n+1}$, so is an affine variety with coordinate ring $k[x_{1}, \dots, x_{n}]_{f}$. **Proof**: Letting $A = k[x_1, \dots, x_{n}]$, the localisation map $A \to A_f$ induces a morphism of affine varieties $\varphi: V(x_{n+1}f-1) \to \mathbb{A}^n$. The preimage of a point $(a_{1},\dots, a_{n}) \in \mathbb{A}^n$ is either empty or is a single point with coordinates $(a_{1}, \dots, a_{n}, f(a_{1}, \dots, a_{n})^{-1})$ in $\mathbb{A}^{n+1},$ so $\varphi$ is injective. In particular, we see that $\text{Im}(\varphi) = H$. Finally, the map $\varphi^{-1}$ is given by $$\varphi^{-1}(a_{1}, \dots, a_{n}) = (a_{1}, \dots, a_{n}, f(a_{1}, \dots, a_{n})^{-1}).$$ Since the coordinate functions are all regular on $V(x_{n+1}f-1)$ we see that $\varphi^{-1}$ is also a morphism of varieties, proving that $H \cong V(x_{n+1}f-1)$. $\square$ **Proposition**: The topology on a variety $Y$ has a base of open affine subsets. **Proof**: It suffices to show that for every open neighbourhood $p \in U$ of a point $p \in Y$, there is an affine open $p \in V \subseteq U$. A subset of $U$ is relatively open if it open in $Y$, so since $U$ is a variety we can reduce to $U=Y$, and since varieties have *quasi*affine covers we can assume $Y \subseteq \mathbb{A}^n$ for some $n.$ Let $Z = \overline{Y} \setminus Y$, and let $\mathfrak{a}$ be the ideal corresponding to this closed subset of $\mathbb{A}^n$. Since $p \not \in Z$, there is some polynomial $f \in \mathfrak{a}$ with $f(p) \neq 0$. We have $Z \subseteq V(f) =: H$, and since $H$ does not contain $p$, $p \in Y \setminus Y\cap H$ open. On the other hand, in $\mathbb{A}^n$ we have $Y \setminus Y \cap H \subseteq \overline{Y}\cap (\mathbb{A}^n \setminus H) \subseteq \mathbb{A}^n \setminus H$ a closed subset of an affine variety, so $Y \setminus Y \cap H$ is itself affine. $\square$ **Theorem**: The category of varieties over $k$ with morphisms the dominant rational maps is equivalent to the category of finitely generated field extensions of $k$. We have a bijective correspondence $$ \bigl\{\text{dominant rational maps } X \to Y\bigr\} \iff\bigl\{ k\text{-algebra homomorphisms } K(Y) \to K(X)\bigr\}. $$ **Proof**: If $\varphi: X \to Y$ is dominant rational represented by its values on an open $U \subseteq X$ with dense image, then for $f \in K(Y)$ regular on $V \subseteq Y$ we have $f\varphi : \varphi|_{U}^{-1}(V) \to k$ regular and $\varphi|_{U}^{-1}(V)$ is nonempty open by density of $\varphi(U)$. This construction makes the assignment $X \mapsto K(X)$ a contravariant functor from varieties with dominant rational maps to the category of field extensions of $k$, $$K\bigl( X \xrightarrow{\varphi} Y \bigr) = \quad \varphi_{*} : K(Y) \to K(X), \ \langle V, f\rangle \mapsto \langle \varphi_{U}^{-1}(V), f\varphi|_{U}\rangle.$$We have $K(Y) = K(U)$ for any open subset $U$ of $Y$ by definition, so by the above proposition every variety has the function field of an affine variety. For affine varieties the function field is just the fraction field of the coordinate ring, which is finitely generated with $\text{tr. deg}_{k}K(U) = \text{dim}(U)$. It follows that the functor $K$ has image contained in the full subcategory of finitely generated extensions of $k$. To show that $K$ is fully faithful we give an inverse construction. Let $\theta : K(Y) \to K(X)$ be a $k$-homomorphism, and assume without loss of generality that $Y$ is affine. The coordinate ring $A(Y)$ of $Y$ is finitely generated, and if we choose generators $y_1, \dots, y_{n}$ then there are open sets $U_i \subseteq Y$ for $y_i$ on which $\theta(y_i)$ is defined, and $\theta(y_i)$ is regular on an open set $V_i$ in $X$. Since the $V_i$ contain distinguished open sets $D(f_i)$ for some polynomial $f_i$ (from the proposition), their intersection $V_1 \cap \dots \cap V_{n} =: V$ contains $D(f_{1}\dots f_{n})$ so is nonempty and open. This means that the map $y_i \to \theta(y_i)$ gives an injective homomorphism $A(Y) \to \mathcal{O}(V)$ (because injective at the function field level), corresponding to a dominant morphism $V \to Y$ of varieties, or in other words a dominant rational map $\theta': X \to Y$. Precomposition with $\theta'$ recovers $\theta$, so this construction inverts $K$ on morphisms. The last thing to check is that $K$ is essentially surjective: every finitely generated extension of $k$ is the function field of some variety. Let $L/k$ be finitely generated, and choose generators $y_1, \dots, y_{n}$. Then the sub-$k$-algebra $B$ generated by the $y_i$ (leaving out their inverses, but keeping relations between them) is a quotient of the polynomial ring $k[x_1, \dots, x_{n}]$, making $B$ the coordinate ring of an affine variety $Y$. It follows that $K(Y) \cong L$. $\square$ With this theorem we have completely answered our question (1): varieties and dominant rational maps is the geometric category corresponding to maps of function fields. We have also made some progress towards (2): the proof shows that every variety is birational to any of its affine open subsets, and on these subsets all regular functions are in the coordinate ring, ie they are polynomials satisfying some relations. For a complete answer we would like a better model of the function field: a class of (affine) varieties whose coordinate rings also have relations we understand. To construct such a model we will use some field theory. **Primitive Element Theorem**: If $L/K$ is a finite separable extension, there is an element $\alpha$ such that $L = k(\alpha)$. If $\beta_{1}, \dots, \beta_{n}$ are generators for $L$ as a $K$-vector space and $K$ is infinite, then we can take $\alpha = c_{1}\beta_{1} + \dots + c_{n}\beta_{n}$ for some $c_1, \dots, c_{n} \in K$. **Definition**: A field extension $K/k$ is **separably generated** if it has a transcendence base $\{x_{i}\}_{i \in I}$ so that $K/k(\{x_{i}\})$ is a separable extension (recall that usually this extension is only assumed to be algebraic). If this is the case, we call $\{x_{i}\}$ a **separating transcendence base**. **Theorem** \[ [Zariski-Samuel Vol I Chp I § 13 Thm 30](https://ia801500.us.archive.org/2/items/in.ernet.dli.2015.134674/2015.134674.Commutative-Algebra-Volume--1.pdf) ]: Let $K/k$ be a separably generated with finite cardinality transcendence base. Then any set of generators for $K$ contains a separating base. The proof of this result (Maclane's theorem) inducts on the number of generators in a separating base. Once the result is known for one generator the inductive step simply involves ordering the variables so that we have a composition of extensions, one with a separating base of one element. The theoretical input to the key one-generator step is background on *perfect fields*. **Definition**: A field $k$ is **perfect** if it is characteristic 0, or if it is characteristic $p$ and $k = k^p$, ie every element has a $p$th root. In characteristic 0, every algebraic extension of a field is separable, and in characteristic $p$ this also holds if the base field is perfect (see the aforementioned chapter of Zariski-Samuel for details of the proof). Note in particular that algebraically closed fields in any characteristic are perfect. This result also extends to transcendental extensions, as we see below. **Theorem** \[ [Zariski-Samuel Vol I Chp I § 13 Thm 31](https://ia801500.us.archive.org/2/items/in.ernet.dli.2015.134674/2015.134674.Commutative-Algebra-Volume--1.pdf) ]: If $k$ is perfect, then any extension $K/k$ with finite transcendence degree (ie a finitely generated extension) is separably generated. **Theorem**: Any variety $X$ of dimension $n$ is birational to a hypersurface $Y$ in $\mathbb{P}^{r+1}$. **Proof**: Since the field $k$ of definition is algebraically closed, the function field $K(X)/k$ is finite separably generated over $k$, so there is a transcendence base $\{x_1, \dots, x_{r}\}$ such that $K(X)/k(x_{1}, \dots, x_{r})$ is finite separable. The primitive element theorem implies that $K(X) = k(x_1, \dots, x_{r}, y)$ for some separable element $y$ over $k(x_{1},\dots, x_{r})$. Since $y$ is a zero for an irreducible rational polynomial in $x_1, \dots, x_{r}$, we can clear denominators to express as a root of an irreducible polynomial $f(x_1,\dots, x_{r}, y)$ in $k[x_{1}, \dots, x_{r}, y]$. The polynomial $f$ cuts out a hypersurface $V(f) \subseteq \mathbb{A}^{r+1}$ satisfying $$K(V(f)) = \text{Frac}\left( k[x_{1},\dots, x_{r},y]/(f) \right) = k(x_{1},\dots, x_{n},y),$$so by the equivalence of categories between varieties with rational maps and function fields we have that $X$ is birational to $V(f)$. The projective closure gives the corresponding hypersurface $Y$. $\square$ **Remark**: It may be unclear why we take the projective closure when we have already obtained a good affine model for the variety. One reason for this is to allow us to use the strongest scheme-theoretic machinery available to study the birational equivalence class of $X$: projective structure has features such as a grading on the homogeneous coordinate ring and the existence of natural ample line bundles on the variety which are helpful for algebraic applications. Topologically, it may be desirable to compactify $V(f)$, and the projective closure is a simple way to do this. We now also have (2): on our affine model for $X$, the regular functions are described by polynomials modulo one relation. The drawback of this statement is that the description is only valid on a dense open subset of $X$, which we have not identified explicitly.

Lüroth's Theorem and Simple Connectedness of P1

Sun, 01 Mar 2026 00:00:00 GMT

import factoredCurves from "../../../images/factored-curves.png" Over the last few days I've been learning some machinery used to analyse $\mathbb{G}_{m}$-actions in algebraic geometry. One of the results I've needed for this is Lüroth's theorem. In this write-up I will follow the proof given in Hartshorne, which makes the geometric content of the statement clear and establishes theory about morphisms of curves along the way. This is a favourite theme in my recent work - analysis of field extensions usually provides strong information about morphisms of varieties. I especially enjoy results in the characteristic $p$ case; the field theory developed on its own can feel overly complicated. The main result from curve theory we use is Hurwitz's theorem. In this note we will assume all curves are nonsingular projective, and make note if results extend more generally. **Definition**: Let $f : X \to Y$ be a morphism of curves. We say $f$ is **separable** if the induced extension $K(X)/K(Y)$ of function fields is separable. **Hurwitz's Theorem**: Let $f: X \to Y$ be a finite separable morphism of curves, with degree $n = [K(X): K(Y)]$. Then $$ 2g(X)-2 = n(2g(Y)-2) + \text{deg}(R), $$ where $R = \sum_{p \in X} \text{length}(\Omega_{X/Y, \ p}) p$ is the **ramification divisor** and $g(X)$ is the (geometric) genus. If $f: X \to Y$ is a finite morphism of curves, then $K(X)/K(Y)$ is a finite extension and there is a subfield $L$ of $K(X)$ such that $K(X)/L$ is purely inseparable and $L/K(Y)$ is separable. By the equivalence of categories between fields of transcendence degree one and function fields of curves, to $L$ corresponds a curve $Z$ for which we have a factorisation

We can use Hurwitz to study the separable part, so we are left to analyse the purely inseparable part $X \xrightarrow{g} Z$ (recall that inseparability is a characteristic $p$ phenomenon, so we can assume $X$ is defined over $\mathbb{Z}_{p}$ for some prime $p$). **Definition**: Let $X$ be a scheme with local rings of characteristic $p$. The **Frobenius morphism** $F: X \to X$ is defined to be the identity map on the topological space of $X$, with corresponding sheaf morphism $F^{\#} : \mathcal{O}_{X} \to \mathcal{O}_{X}$ the $p$th power map. The local rings assumption ensures $F$ is a morphism. For a ring $A$, the map $$A \to A, a \mapsto a^p$$ is an additive homomorphism if and only if all middle terms of $(a+b)^p$ vanish for every choice of $a$ and $b$, which holds for every characteristic $p$ ring. If $X$ is a $k$-scheme and $k$ has characteristic $p$, the Frobenius morphism is not $k$-linear, instead satisfying $\pi F = F \pi$, where $\pi$ is the structure morphism $X \xrightarrow{\pi} \text{Spec}(k)$. This can be thought of as $k$-linearity for maps between two different $k$-schemes: let $X_p$ have the same scheme structure as $X$, but with structure morphism $\pi' = F \pi : X_{p} \to \text{Spec}(k)$. Then our equation reads $\pi' = \pi F$, so that the Frobenius *is* a $k$-linear morphism $F' : X_{p} \to X$. To distinguish the two perspectives we call $F'$ the **$k$-linear Frobenius morphism**. **Proposition**: Let $f: X \to Y$ be a finite morphism of curves such that $K(X)/K(Y)$ is a purely inseparable extension. Assume $X$ and $Y$ are defined over an algebraically closed field $k$ of characteristic $p$. Then $X$ and $Y$ are isomorphic as abstract schemes (excluding the structure morphisms) and $f$ is a composition of $k$-linear Frobenius morphisms. In particular, $g(X) = g(Y)$. **Proof**: The $k$-linear Frobenius map $Y_{p} \to Y$ corresponds on function fields to the injection $K(Y) \xhookrightarrow{\hat{} \, p} K(Y)$, which is a degree $p$ extension $K(Y)/K(Y)^p$. The extension is equivalent to the extension $K(Y)^{1/p}/K(Y)$ where the top field is obtained by adding all $p$th roots in an algebraic closure of $K(Y)$ to $K(Y)$, in the sense that there is an isomorphism $K(Y) \cong K(Y)^{1/p}$ which sends $K(Y)^p$ to $K(Y)$. Now if $K(X)/K(Y)$ is finite and purely inseparable in characteristic $p$, its degree is $p^r$ for some $p$. Since $K(X)^{p^r} \subseteq K(Y)$ we have $K(X) \subseteq K(Y)^{1/p^r}$, and since $K(X)/K(Y)$ and $K(Y)^{1/p^r}/K(Y)$ both have degree $p^r$ we see that $K(X) = K(Y)^{1/p^r}$. The sequence of curve morphisms $$Y_{p^r} \xrightarrow{F'} Y_{p^{r-1}} \xrightarrow{F'} \dots \xrightarrow{F'} Y_{p} \xrightarrow{F'} Y$$ with $Y_{p^r} = (Y_{p^{r-1}})_{p}$ for $r \geq 1$ realises $K(Y)^{1/p^r}$ as the function field of a finite purely inseparable morphism $Y_{p^r} \xrightarrow{(F')^r} Y$, so since nonsingular projective curves are determined up to isomorphism by their function fields we see that $X \cong Y_{p^r}$. $\square$ The moral of this result is that the interesting finite morphisms of curves are exactly the separable ones (the inseparable part has a standard form). This means that Hurwitz's theorem is essentially a universal tool for studying these morphisms. **Lemma**: Let $f : X \to Y$ be a finite morphism of curves, $g(Y) \geq 1$. Then $g(X) \geq g(Y)$. **Proof**: As discussed above, we can factor into a separable and a purely inseparable part, and genus is unchanged for the purely inseparable part of the extension. Assume without loss of generality that $f$ is separable. Then we can rearrange the formula from Hurwitz's formula: $$ \begin{aligned} 2&g(X) - 2 = n(2g(Y) -2) + \text{deg}(R) \\ \implies &g(X) = n(g(Y)-1) + \frac{1}{2}\text{deg}(R) + 1 \\ = \ &g(Y) + (n-1)(g(Y)-1) + \frac{1}{2}\text{deg}(R). \end{aligned} $$ In the right-hand side, the ramification divisor has non-negative degree and $n, g(Y) \geq 1$, so we see that $g(X) \geq g(Y)$. $\square$ Using the formula in the lemma, we have $g(X) = g(Y)$ if and only if $\text{deg}(R) = 0$ and either $n =1$ or $g(Y) = 1$. In other words, there are no finite morphisms between non-isomorphic curves of genus $\geq 2.$ We have yet to cover the genus 0 case, which we handle separately since we can say more. **Definition**: A curve $Y$ is **simply connected** if for every finite étale morphism $f: X \to Y$, $X$ is isomorphic to the disjoint union of $\text{deg}(f)$ copies of $Y$. We say $Y$ has **no nontrivial étale covers**. **Remark**: The Frobenius morphism (and thus any inseparable extension) is not an étale cover. In fact, $F'$ is everywhere ramified: since $F'$ induces the $p$th power map on local rings, if $Q \in Y$ is any point and $t$ a local parameter at $Q$ we have $F^{'\small\#}t = t^p$ with valuation $p$ in $\mathcal{O}_{Y, Q}$. **Lemma**\[[HS IV-1.3.5](https://link.springer.com/book/10.1007/978-1-4757-3849-0)]: Let $X$ be a (nonsingular projective) curve. Then the following are equivalent:

(a) $X$ is rational
(b) $g(X) = 0$
(c) $X$ is isomorphic to $\mathbb{P}^1$.

**Lemma**: $\mathbb{P}^1_{k}$ is simply connected. **Proof:** Let $f : X \to \mathbb{P}^1_{k}$ be a finite étale morphism, and assume $X$ is connected. Then $X$ is smooth over $k$ (composition of smooth morphisms is smooth) and proper because $f$ is finite, which makes $X$ a curve. A finite étale morphism of curves is separable, so by Hurwitz's theorem $$2g(X) - 2= -2n.$$ Since $n \geq 1$ and $g(X) \geq 0$, this identity holds if and only if $n =2$, $g(X) = 0$. By the lemma above, we conclude that $X \cong \mathbb{P}^1$. $\square$ **Lüroth's Theorem**: Let $k$ be an algebraically closed field. If $k(t)/k$ is a purely transcendental extension of degree 1, any subfield $L$ is also purely transcendental over $k$. **Proof**: Assume that $L \neq k$, so that $L$ has transcendence degree 1. Then $L$ is the function field of a curve, and the extension $k(t)/L$ is finite (or else $k(t)$ would have transcendence degree $\geq 2$) corresponding to a finite morphism $f: \mathbb{P}^1_{k} \to Y$ ($K(Y) = L$). By the genus inequality $Y$ cannot have genus $g(Y) \geq 1$, so $g(Y) = 0$ and $Y \cong \mathbb{P}^1$. It follows that $Y \cong k(u)$ for some $u$. $\square$

Einstein Summation

Sat, 28 Feb 2026 00:00:00 GMT

I often have trouble remembering the conventions for summation and which spaces the vectors are supposed to live in. I would really like to have good notes on graduate differential geometry as a whole in the future, but for now I will start with something I need to use a lot. **Definition**: Let $V$ be a finite-dimensional (real or complex) vector space with dual $V^*$. The space $$V^r_{s} = V^{\otimes r} \otimes V^{* \ \otimes s}$$ is called the space of **type (r,s)-tensors**. The integer $r$ is called the **contravariant order** and $s$ is the **covariant order**. **Contravariance** here means that under a change of basis $P_{\mathcal{B} \mathcal{B'}}$ from basis $\mathcal{B}$ to $\mathcal{B'}$, the coordinate vector $\begin{bmatrix}v_{1} \\ \vdots \\ v_{n}\end{bmatrix}_{\mathcal{B}}$ of any point $v \in V$ , if interpreted as $\mathcal{B'}$ coordinates instead, gives coordinates for the point $P_{\mathcal{B} \mathcal{B'}}^{-1}v$. On the other hand, **covariance** says that with change-of-basis $P_{\mathcal{B} \mathcal{B'}}$ for $V$, the coordinates $\begin{bmatrix}v^*_{1} \\ \vdots \\ v^*_{n}\end{bmatrix}_{\mathcal{B}}$ for vector $v^* \in V$ with respect to $\mathcal{B}$ correspond to $P_{\mathcal{B} \mathcal{B'}}^Tv^*$ as coordinates for $\mathcal{B'}$ (since the dual transformation to left-multiplication by $P_{\mathcal{B} \mathcal{B'}}$ is given by left-multiplication by $(P_{\mathcal{B} \mathcal{B'}}^{-1})^T$. Physicists also like to define tensors as collections of coordinates specified by type $(r,s)$-multi-indices and subject to the co/contravariance relations, which is an equivalent formulation. For calculations, we use **row vectors** for coordinates in $V^*$ and **column vectors** for coordinates in $V$. **Tensor contraction**: For any pair of indices $1 \leq \lambda \leq r$, $1 \leq \mu \leq s$ we get a homomorphism $C_{\lambda \mu} : V^r_{s} \to V^{r-1}_{s-1}$, defined by extending the map $$\begin{aligned} &C_{\lambda \mu}(v_{1}\otimes{\dots} \otimes v_{r}\otimes v^{* \ 1}\otimes{\dots} \otimes v^{* \ n}) \\ = \ &\langle v_{\lambda}, v^{* \ \mu}\rangle v_{1}\otimes\dots \otimes \hat{v}_{\lambda} \otimes \dots \otimes v_{r}\otimes v^{* \ 1}\otimes\dots \otimes \hat{v}^{* \ \mu} \otimes \dots \otimes v^{* \ n}\end{aligned}$$ to linearity (where $\hat{}$ means to remove those indices from the tensor). **Raising and lowering indices with a (pseudo)metric**: If instead of the canonical pairing between $V$ and $V^*$ we contract using an inner product with matrix $\eta_{\mu \nu}$ then we can write the front coefficient either as $v_{\lambda}$

The Artin-Schreier Theorem

Thu, 26 Feb 2026 00:00:00 GMT

I would like this note to be a running document on Galois theory in characteristic $p$ and mixed characteristic for algebraic geometry. Hopefully the scope will become apparent in time! The last few weeks I've slowly prepared to learn the Artin-Schreier Theorem, a project which was originally motivated by wanting to attend lectures on perfectoid spaces and almost rings. As the weeks of linear algebra tutoring went by it became clear to me that I was unhappy attending without more background, so my goal has instead become to understand this theorem and its application in that context. Now that I've finally achieved part of that, I think the goal is to learn some Kummer theory and the Witt vector version of the story, and then I will see how I can course-correct from there. From my current perspective, it looks like the goal of this theory is to explicitly describe finite cyclic extensions of fields. We use two powerful theorems in field theory to facilitate the analysis: 1. **Hilbert's Theorem 90**: Let $K/k$ cyclic of degree $n$. An element $\beta \in K$ has norm $N^K_{k}(\beta) = 1$ if and only if there is $\alpha \in K^\times$ with $\beta = \frac{\alpha}{\sigma(\alpha)},$ with $\sigma$ a generator of $\text{Gal}(K/k).$ An element $\beta \in K$ has trace $\text{Tr}^K_{k}(\beta) = 0$ if and only if there is $\alpha \in K$ such that $\beta = \alpha - \sigma(\alpha)$. 2. **Artin's Linear Independence of Characters**: Let $\chi_1, \dots, \chi_n : K^\times \to K^\times$ be distinct homomorphisms, $K$ a field. Then the $\chi_i$ are $K$-linearly independent. We first analyse the simplest case. **Theorem**: Let $k$ be a field, and let $n \in \mathbb{Z}_{> 0}$ be prime to the characteristic of $k$. Assume that $k$ contains a primitive $n$th root of unity. Then: 1. Cyclic extensions $K/k$ of degree $n$ admit primitive elements $\alpha$ which satisfy a polynomial $X^n - a$ for some $a \in k$. 2. For a polynomial $X^n -a$ and a root $\alpha$ in the algebraic closure of $k,$ the extension $k(\alpha)/k$ is cyclic of degree $d \ | \ n$ and we have $\alpha^d \in k$. **Proof**: If $K/k$ is cyclic of degree $n$, then for $\zeta$ a primitive $n$th root in $k$ we have $N^K_k(\zeta^{-1}) =1,$ so by Hilbert 90 there is some $\alpha \in K$ with $\sigma(\alpha) = \zeta \alpha \leftrightarrow \zeta^{-1} = \frac{\alpha}{\sigma(\alpha)}$. The Galois conjugates of $\alpha$ are then $\alpha, \zeta\alpha, \dots, \zeta^{n-1}\alpha$ (all distinct), so that $k(\alpha) = K$. We have $\sigma(\alpha^n) = \sigma(\alpha)^n = \alpha^n$, so that $\alpha^n$, being fixed by the action of the Galois group, is in $k$ and $\alpha$ is a root of $X^n - (\alpha^n)$. If $\alpha$ is a root of $X^n - a$ then so is $\zeta^i \alpha$ for $i=1, \dots, n$, which makes $k(\alpha)/k$ a normal extension. Because the roots are distinct, the extension is also separable, hence Galois. Let $\sigma$ generate $\text{Gal}(k(\alpha)/k)$, and write $\sigma(\alpha) = \omega \alpha$, $\omega$ a primitive $d$th root of unity for some $d \ | \ n$. Then $\sigma(\alpha)^d =\omega^d \alpha^d = \alpha^d$, so that $\alpha^d \in k$ and $k(\alpha)/k$ is cyclic of order $d$. $\square$ The Artin-Schreier theorem uses the same techniques as the above theorem but with the additive version of Hilbert 90. Note that here the characteristic plays the role that existence of a $n$th root of unity did in the multiplicative case. **Artin-Schreier Theorem**: Let $k$ be a field of characteristic $p$. Then: 1. Cyclic extensions $K/k$ of degree $p$ admit primitive elements $\alpha$ satisfying a polynomial $X^n - X - a$ for some $a \in k$. 2. A polynomial $X^n - X -a, \ a \in k$ either has no roots in $k$ or *all* roots in $k$. In the former case, the polynomial is irreducible and for any root $\alpha$, $k(\alpha)/k$ is cyclic of order $p$. **Proof**: Let $K/k$ be cyclic of degree $p$. Then we have $\text{Tr}^K_{k}(-1) = 0$, so by additive Hilbert 90 there is $\alpha \in K$ with $\sigma(\alpha) = \alpha +1$. The elements $\alpha, \alpha + 1, \dots, \alpha + (p-1)$ are the distinct conjugates of $\alpha$ in $K$ and there are $p$ of them, so $K = k(\alpha)$. We have $$\sigma(\alpha^p) = \sigma(\alpha)^p = (\alpha+1)^p = \alpha^p + 1 (\text{binomial theorem mod $p$}),$$ so that $\sigma(\alpha^p - \alpha) = \alpha^p - \alpha =: a \in k$ and $\alpha$ is a root of $X^n - X - a$. Now consider the polynomial $f(X) = X^n - X -a$ with $a \in k$. If $\alpha \in k$ is a root of $f(X)$, then by what we have shown above also $\alpha + 1, \dots, \alpha + (p - 1)$ are roots in $k$ and the polynomial splits in $k$ as soon as it has a single root in $k$. Otherwise, assume there are no roots in $k$. To show irreducibility, suppose that $f(X) = g(X)h(X)$ where $1 \leq \text{deg}(g) < p$. Then $g$ is a product of $d$ factors $X - \alpha - i$ for distinct choices of $i$ (we know how $f$ splits in $k(\alpha)$), and in particular the coefficient of $X^{d-1}$ in $g$ has the form $d\alpha + j$ for some integer $j$ mod $p$. But we have $g \in k[X]$, so $d\alpha + j \in k$ with $j \in k, d \in K^\times$ tells us that also $\alpha \in k$, contradicting our assumption on roots of $f$. So $f(X)$ is irreducible with $p$ distinct roots in $k(\alpha)$, making $k(\alpha)/k$ Galois. Since $\alpha, \alpha + 1$ are both roots of $f$ there is some $\sigma \in \text{Gal}(k(\alpha)/k)$ with $\sigma(\alpha) = \alpha+1$, so that $\text{Gal}(k(\alpha)/k)$ is cyclic with $\sigma$ a generator. $\square$

Calculating normal cones

Thu, 26 Feb 2026 00:00:00 GMT

import planeline from "../../../images/plane-line-colour-grading.jpg" One of the more frustrating problems in day-to-day ring theory is determining isomorphisms of rings. In algebraic geometry it is often the case that problems involve adding or removing variables subject to equations to polynomial rings (geometric content: going up and down in dimension, which we do for blowups, vector bundles, covers, cones, etc), and there is no silver bullet for finding the simplest description of such rings. To the beginner in algebraic geometry it is difficult to know what options we have for speeding up these calculations, either through existing algorithms or variants we construct ourselves. Statements which appear obvious can turn out to be difficult, even equipped with the usual tricks like the Nullstellensatz. Today I spent a few hours solving one of these problems, aiming to reinforce my improvisation skills on the problem and develop some new techniques. I found this problem while preparing to study virtual fundamental classes, which requires some familiarity with intersection theory. This lead me to learning about normal cones and (soon) obstruction theory. As a first step to this, I am looking at the normal cone to the projective variety $X = V(xz, yz) \subseteq \mathbb{P}^3_{[x:y:z:w]}$ (below: a picture of $X$ in the affine chart $w \neq 0$).

The scheme-theoretic definition of the normal cone is as a relative spectrum, defined by gluing together spectra over the affine open sets in $X$, so to understand the cone we can study it over the chart $U_w = \{w \neq 0\}$. We have $X \cap U_w = V(xz, yz)$ (this time, the affine vanishing locus) given by ideal $I = (xz, yz)$, so that the normal cone has description $C_{X/\mathbb{P}^3}|_{U_w} = \text{Spec}(\bigoplus_{n\geq 0}I^n/I^{n+1})$. Writing $R = \mathbb{C}[x, y, z]/(xz, yz)$ for the coordinate ring of $X\cap U_w$, I wanted to show that $\bigoplus_{n\geq 0} I^n/I^{n+1} \cong R[A, B] /(yA - xB).$ The general technique I used is an inductive argument on relations. Since the map $$ \begin{aligned} \varphi: R[A, B] &\to \bigoplus_{n\geq 0} I^n/I^{n+1}, \\ \quad A \mapsto xz&, \quad B \mapsto yz \end{aligned} $$ is a surjective graded homomorphism, the kernel is a homogeneous ideal, so we need to show that for every degree $n$ relation $S = \sum_{i=0}^{n} r_{i} A^{n-i}B^i,$ $$ \varphi(S) = 0 \implies S \in (yA - xB). $$ In this example, the variable $z$ is irrelevant to the relation being 0 because $\varphi(r_iA^{n-i}B^i)$ is always divisible by $z^n$: we have $\varphi(A^{n-i}B^i) = z^n x^{n-i}y^i$, so to cancel terms each $r_i$ will need to be divisible by either $x$ or $y$. If $z$ divides $r_i$ as well, the term was already zero in $R[A, B]$. Returning to generalities, a precise formulation of the approach we use is the following:

Deduce some divisibility statement on the coefficient $r_i$.
Show that this information gives us a relation $S'$ with the same image in $\bigoplus_{n\geq 0} I^n/I^{n+1}$ and which is reducible; induction says that $S'$ lives in the target ideal.
Look at the difference $S - S'$; if this lives in the target ideal then we see that $S$ does as well.

Of course, there is a little yoga to this: we need to choose $S'$ so that step 3 will happen in the way we expect. For ideals generated by one linear homogeneous polynomial this isn't too difficult, but the reasoning could get more complicated with more variables and in more degrees. In our set-up, (1) is easy: we need to have $y \ | \ r_0$ and $x \ | \ r_{n}$ in order for these terms to appear and be nonzero in $S$, because all terms other than $A^n$ have image divisible by $y$ and all terms other than $B^n$ have image divisible by $x$. This means that
$S' = \frac{xr_{0}}{y}\cdot BA^{n-1} + \sum_{i=1}^n r_{i} A^{n-i}B^i$ has the same image and factorises as $$ B\left( \frac{x r_{0}}{y} A^{n-1} + \sum_{i=1}^n r_{i}A^{n-i}B^{i-1} \right), $$ which maps to 0 if and only if the bracketed term does. This gives us step (2). For step (3) the difference is $$S - S' = r_{0}A^n - \frac{xr_{0}}{y}A^{n-1}B,$$ and taking out the factor $\frac{r_{0}}{y}A^{n-1}$ leaves us with $yA - xB$. As an **exercise**, one can check that the same holds for the affine normal cone over $V(x^2, xy) \subseteq \mathbb{A}^2$: with $R = \mathbb{C}[x,y]/(x^2, xy)$ we have
$\bigoplus_{n\geq 0} I^n/I^{n+1} \cong R[A, B]/(yA - xB)$ using the surjective graded homomorphism of $R$-algebras $A \mapsto x^2, B \mapsto xy$.

Group Varieties and Schemes

Tue, 24 Feb 2026 00:00:00 GMT

Yesterday I spent a little time on a Hartshorne exercise which checks that $\mathbb{G}_{a}$ and $\mathbb{G}_{m}$ are group schemes. By now I've thought about this a few times and it's included in my thesis, but I learnt some new ideas by proving this in the category of varieties, which feels like another piece of working with the variety-theoretic definition of regular functions (something I've always been uneasy about, I gravitate to the scheme theory because I can use more algebra). Elements of the ground field $k$ correspond to points/maximal ideals of $\mathbb{G}_{a}$ one-to-one and elements of $k^* = k \setminus \{0\}$ correspond to points of $\mathbb{G}_{m}$, so that functions $X \to k$ from a variety correspond to functions $X \to \mathbb{G}_{a}$ and functions $X \to k^*$ correspond to functions $X \to \mathbb{G}_{m}$. It follows that the function $X \to k$ is regular if and only if $X \to \mathbb{G}_{a}$ is a morphism, and the same holds in the $\mathbb{G}_{m}$ case. Having such an explicit correspondence is probably a unique feature of (products of) $\mathbb{G}_{m}$ and $\mathbb{G}_{a}$, but checking representability of a moduli functor by looking at $k$ points is a useful idea, made simpler by the regular function notion. I also reinforced for myself the idea that sometimes the best way is to compute and find out. I was trying to write down the comultiplication for $\mathbb{G}_{m}$ and $\mathbb{G}_{a}$ and my immediate instinct was to google to make sure I didn't start something monstrous and complicated. I did google and my guess was correct (it was, after all, an educated guess), but it turns out that checking the induced map on points is actually very doable, and this would have been true even with the wrong guess. For example, the comultiplication for $\mathbb{G}_{m}$ is given by $$k[x^{\pm 1}] \xrightarrow{\mu} k[x^{\pm 1}, y^{\pm 1}], \quad x \mapsto xy;$$ since $\mu^{-1}(x-a, y-b)$ has form $(x-c)$ for some $c \in k$ and $\mu(x-c) = xy - c$, we find that the only element of $(x-a, y-b)$ with $1$ for the coefficient of $xy$ and no other terms is $$(x-a)(y-b) + b(x-a) + a(y-b) = xy +ab -ab -ab = xy - ab,$$ so that $c = ab$ and the map on points is multiplication $(a, b) \to ab$.

Symplectic Structure on Varieties

Mon, 23 Feb 2026 00:00:00 GMT

Let $X$ be a variety. A **symplectic structure** on a holomorphic manifold/algebraic variety is a nondegenerate 2-form $\omega \in \Gamma(X, \Omega^2_{X})$ (where $\Omega^2_X$ is the sheaf of holomorphic or algebraic 2-forms on $X$) satisfying $d\omega = 0$. **Example**: Let $M$ be a manifold, $X = T^*M$ its cotangent bundle. We construct a symplectic form $\omega$ as $d\lambda$ for some $\lambda : TX \to \mathbb{C}$, which ensures $d\omega = 0$.
Let $x \in M$ and $\alpha \in X_x$, the fibre over $x$. The projection map $\pi : X \to M$ induces a tangent map $\pi_{*} : T_\alpha X \to T_x M$ , and we define $\lambda$ by setting $$ \lambda(\xi) = \alpha(\pi_{*}\xi), \text{where } \xi \in T_{\alpha}(X). $$ In this description we can see naturality of $\lambda$, but for nondegeneracy and understanding how to evaluate the form an expression in coordinates is preferable. Let $q_1, \dots, q_{n}$ be local coordinates on $M$, and let $p_1, \dots, p_{n}$ be the corresponding dual coordinates on $T^*M$, so that $p_{i}\left( \frac{\partial}{\partial q_{i}} \right) = \delta_{ij}$ and $(q_{1}, \dots, q_{n}, p_{1}, \dots, p_{n})$ gives coordinates for $X$. Then elements of the tangent space $TX$ have the form $$ \xi = \sum_{i=1}^n b_{i}\frac{\partial}{\partial q_{i}}(\alpha, x) + c_{i}\frac{\partial}{\partial p_{i}}(\alpha, x) $$ with $b_i, c_i \in \mathbb{C}$ and $(x, \alpha) \in X$ (so $x \in M, \alpha \in T^*_xM$). The map $\pi$ projects onto the $q$-coordinates, so the corresponding tangent map does the same: $$ \pi_{*}\left( \sum_{i=1}^n b_{i}\frac{\partial}{\partial q_{i}}(x, \alpha) + c_{i}\frac{\partial}{\partial p_{i}}(x, \alpha) \right) = \sum_{i=1}^n b_{i}\frac{\partial}{\partial q_{i}}(x, \alpha). $$ Now we can calculate $$ \begin{aligned} \lambda(\xi) = \alpha\left( \sum_{i=1}^n b_{i}\frac{\partial}{\partial q_{i}}(x, \alpha) \right) &= \sum_{i=1}^n p_{i}(\alpha)p_{i}\left( \sum_{j=1}^n b_{i}\frac{\partial }{\partial q_{i}}(x, \alpha) \right) \\ &= \sum_{i=1}^n b_{i}p_{i}(\alpha) \end{aligned} $$ using our dual coordinates, which identifies $\lambda$ with $\sum_{i} p_{i} \ \mathrm{d} q_{i}$ and hence $\omega$ with $\sum_{i} \mathrm{d}p_{i} \wedge \mathrm{d}q_{i}.$ Now $X$, being a holomorphic manifold, is locally isomorphic to $\mathbb{C}^{2n}$ and the local form of $\omega$ is nondegenerate here: $$ \omega(e_i, e_j) = \frac{1}{2} \sum_{i}\mathrm{d}p_{i}(e_{i})\wedge \mathrm{d}q_{i}(e_{j}) = \begin{cases}1, \ j = i + n \\ -1, \ i = j + n\\ 0, \text{ else} \end{cases}, $$ in the standard basis for $T^*\mathbb{C}^{2n}$, giving block matrix form $$ \omega = \begin{pmatrix} 0 & I_{n} \\ -I_{n} & 0 \end{pmatrix}. $$ Hence we have a symplectic form on $T^*M$ for for any holomorphic manifold, agreeing in local coordinates with the canonical symplectic form on $\mathbb{C}^{2n}$.

A Non-Affine Quasiaffine Variety

Thu, 19 Feb 2026 00:00:00 GMT

To an algebraist the simplest geometric spaces are affine varieties, determined by their ring of functions. Unfortunately, students in algebraic geometry discover early that subvarieties are (a) interesting to study since functions and sheaves restrict to them and (b) indistinguishable by their function rings alone. Using (b) to detect non-affineness, we can show rigorously that the quasiaffine variety $$\Bbb{A}^2 \setminus \{(0,0)\}$$ is not affine. $$\underline{\text{Identifying } \mathcal{O}(\mathbb{A}^2 \setminus \{(0,0)\})}$$: We have an injection $$k[x,y] \to \mathcal{O}(\mathbb{A}^2 \setminus \{(0,0)\})$$ by restriction of regular functions on $$\mathbb{A}^2$$: if $$f, g$$ are regular functions on $$\mathbb{A}^2$$, then $$f-g : \mathbb{A}^2 \to k$$ is also regular and in particular continuous. If $$f, g$$ are identical except possibly at $$(0,0)$$, then since $$f-g$$ is nonzero at $$(0,0)$$ if it is nonzero in an open neighbourhood of $$(0,0)$$ we see that $$f-g$$ must be identically $$0$$. Hence the restriction map is injective. We can also determine the regular functions sheaf-theoretically by using the open cover $$D(x)\cup D(y)$$ for $$\mathbb{A}^2 \setminus \{(0,0)\}$$. These sets contain the same points they would contain in $$\mathbb{A}^2$$, and we know that $$\mathcal{O}(D(x)) = k[x,y]_{x} = k[x^{\pm 1},y]$$, $$\mathcal{O}(D(y)) = k[x, y^{\pm 1}]$$. A regular function on $$\mathbb{A}^2 \setminus \{(0,0)\}$$ is then given by a pair of $$f \in k[x^{\pm1}, y], g \in k[x, y^{\pm 1}]$$ so that $$f$$ and $$g$$ have the same image in $$\mathcal{O}(D(xy)) = k[x, y]_{xy} = k[x^{\pm 1}, y^{\pm 1}]$$. The image of $$f$$ is $$\frac{f}{1}$$ and that of $$g$$ is $$\frac{g}{1}$$, so this means that there is some $$h$$ in $$(xy)$$ with $$h(f-g) = 0 \implies f =g$$ as $$k[x, y]$$ is a domain. Clearly we have $$\mathcal{O}(D(x)) \cap \mathcal{O}(D(y)) = k[x,y]$$ (happening in the function field $$k(x, y)$$), so we see that $$\mathcal{O}(\mathbb{A}^2 \setminus \{(0,0)\}) = k[x,y]$$. Now when $$X$$ and $$Y$$ are varieties and $$Y$$ is affine there is an isomorphism $$\text{Hom}(X, Y) \cong \text{Hom}(A(Y), \mathcal{O}(X))$$. This means that for any affine $Y$ with $$A(Y) \cong k[x,y]$$ one has $$Y \cong \mathbb{A}^2$$, so to determine the difference between $\Bbb{A}^2$ and $\Bbb{A}^2 \setminus \{(0,0)\}$ using maps of rings we will need to look at maps *from* $\Bbb{A}^2 \setminus \{(0,0)\}$ to $\Bbb{A}^2$. Alternatively we could compare their topologies, but this turns out to be surprisingly delicate: a result [Wie78](https://doi.org/10.1112%2Fjlms%2Fs2-18.1.28) of Wiegand shows that all nonempty open subsets of $$\mathbb{A}^2$$ are homemorphic in positive characteristic. $$\underline{\mathbb{A}^2 \setminus \{(0,0)\} \text{ is not isomorphic to } \mathbb{A}^2}$$: Since a morphism of varieties induces a map of the rings of regular functions, an isomorphism $$f: \mathbb{A}^2 \xrightarrow{\sim} \mathbb{A}^2 \setminus \{(0,0)\}$$ would correspond to an automorphism of $$k[x,y]$$. Let $$f_{*}: k[x,y] \to k[x,y], g \mapsto g \circ f$$ be the associated isomorphism of rings (thinking of $$k[x,y]$$ as regular functions on each space), and notice that $$f$$ induces an isomorphism $$k \cong k[x,y]/(x,y) \to k[x, y]/(x \circ f, y \circ f)$$. It follows that $$(x\circ f, y \circ f)$$ is a maximal ideal, so $$(x\circ f, y\circ f) = (x-a, y-b)$$ for some $$a, b \in k$$. Now if $$f : X \to Y$$ is a morphism of varieties and $$g \in \mathcal{O}(Y)$$, then $$V(g)$$ is closed in $$Y$$ and $$V(g\circ f)$$ is closed in $$X$$, $$f(V(g\circ f)) \subseteq V(g)$$ . If $$f$$ is an isomorphism, then applying this reasoning in the other direction gives $$f(V(g\circ f)) = V(g)$$. In our situation, this implies that $$f(V(x-a, y - b)) = V(x, y) = \varnothing$$, which is a contradiction as $$V(x-a, y-b)$$ is a point of $$\mathbb{A}^2$$ for all $$a, b \in k$$. We see that $$\mathbb{A}^2 \setminus \{(0,0)\}$$ cannot be isomorphic to $$\mathbb{A}^2$$. Note that our argument identifying the regular functions on $$\mathbb{A}^2 \setminus \{(0,0)\}$$ will work similarly for $$\mathbb{A}^n \setminus \{(0,0)\}$$ with $$n \geq 3$$. In the $$n = 1$$ case, $$\mathbb{A}^1 \setminus \{0\}$$ *is* a variety, and in this setting we notice that $$\mathbb{A}^1 \setminus \{0\} = D(x)$$ with functions $$k[x]_x$$. **Further reading**: * [BFH19](https://arxiv.org/pdf/1609.06682) for more study of the topology of $$\mathbb{A}^2$$ via subvarieties * [MSEAffineJacobian](https://math.stackexchange.com/questions/47356/are-n-by-n-matrices-with-rank-k-an-affine-algebraic-variety) for relating this result to algebraic Hartog's theorem

Derived Categories and Enumerative Invariants of Genus-One Fibrations

Tue, 17 Feb 2026 00:00:00 GMT

import * as images from "../../../images/scheidegger" ### Introduction In mirror symmetry, one of the ways we approach the motivating conjectures (homological projective duality and general existence of mirrors for Calabi-Yau threefolds) is by collecting evidence, ie writing down invariants of Calabi-Yaus and their mirrors and finding different ways to calculate and compare them. For $X$ a simply connected Calabi-Yau threefold, there are a few types of invariants we are interested in: - Diffeomorphism invariants - Hodge numbers, especially $h^{1,1}(X) = \text{rk} \ H^2(X, \mathbb{Z})$ and $h^{2,1}(X) = \text{rk} \ H^3(X, \mathbb{Z})$ - Invariants under change of Kähler structure - derived category of coherent sheaves $\mathsf{D}^{\flat}(\text{Coh}(X))$ - Deformation invariants - Gromov-Witten, Donaldson-Thomas invariants ### Examples Here are some examples of spaces with agreeing invariants: 1. Let $V$ be a 7-dimensional complex space, and consider the Grassmannian $\text{Gr}_2(V) \subseteq \mathbb{P}(\bigwedge^2 V).$ If $H$ is a hyperplane class in the Grassmannian, the space $X = \text{Gr}_2(V)\cap H^7$ has $K_X = \mathcal{O}_X$, $$h^{1,1} = 1, h^{2,1} = 50, H^{3} = 3, c_{2}\cdot H = 84$$ (where $H^3 = 3H$ in the cohomology/Chow ring of $X$ and $c_2$ is the second Chern class). Let $\varphi : \mathbb{P}(V) \to \mathbb{P}(V^\vee)$ with $\varphi = -\varphi^\vee$ . Let $Y = D_4(\varphi) = \{x \ | \ \text{rk} \phi(x) \leq 4\}$. Then $X$ and $Y$ are not birational, but nonetheless $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(Y))$. 2. (Reye congruence) Let $V$ be a 5-dimensional complex space, and let $\mathscr{X}$ be the Chow variety of two points in $\mathbb{P}(\text{Sym}^2(V))$. The image of the canonical morphism $\mathbb{P}(V) \xhookrightarrow{} \mathbb{P}(\text{Sym}^2(V))$ is a 4-plane determined by a linear system of 5 quadrics $|Q_1\dots Q_{5}|$, and if $\mathscr{H}$ is the locus of singular quadrics one has $\mathsf{D}^{\flat}(\text{Coh}(\mathscr{X})) = \mathsf{D}^{\flat}(\text{Coh}(\mathscr{H}))$. 3. (Octic double solids) Let $X$ be a smooth complete intersection in $\mathbb{P}^7$, with invariants $$ h^{1,1} = 1, h^{2,1} = 65, H^3 = 16, c_{2} \cdot H = 64. $$ Let $Y \xrightarrow{2:1} \mathbb{P}^3$ be a double cover branched over $$ B = \{x \in \mathbb{P}^3 \ | \ \text{det}(A(x)) = 0\}, $$ where $A(x)$ is a matrix associated to $x$ in $M_3(\mathbb{C}[x])$. Then $Y$ is a singular Calabi-Yau threefold having 84 nodal singularities. $Y$ admits a noncompact crepant resolution to a projective space $(\mathbb{P}^3, \mathcal{B})$ equipped with a sheaf of Azumaya algebras, and $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(\mathbb{P}^3, \mathcal{B}))$. For another equivalence, $Y$ has an analytic crepant small resolution to a space $\hat{Y}$ with $\text{Br}(\hat{Y}) \cong \mathbb{Z} / 2\mathbb{Z}$, and with $\alpha$ the nontrivial class in the Brauer group one has $\mathsf{D}^{\flat}(\text{Coh}(X)) = \mathsf{D}^{\flat}(\text{Coh}(\hat{Y}, \alpha)).$ These examples are analysed in the literature using the GLSM formalism: Example 1 is in [Hori Tong '07](https://arxiv.org/pdf/hep-th/0609032) Example 2 is in [Hori '13](https://arxiv.org/pdf/1104.2853) Example 3 is in [Calderaru et al '07](https://arxiv.org/pdf/0709.3855) Another famous example is the Orlov space, this in [Hori Herbst Page '08](https://arxiv.org/pdf/0803.2045) A feature all of these examples have in common is that $h^{1,1} = 1$, how can we get examples with $h^{1,1} = 2$? $\to$ Genus one fibrations. ### The GLSM formalism A gauged linear sigma model (GLSM) is a 5-tuple of data $(G, \rho, R, W, t)$, where: - $G$ is a compact Lie group - $\rho : G \to V$ is a faithful unitary representation of $G$ - $R$ is a faithful unitary representation of $\mathsf{U}(1)$ (same target space as $\rho$) - $W$ is an element of the $G$-invariants ring $S^G$ of $S = \mathsf{Sym}^\bullet \ V^\vee$ with $R$-weight 2 - $t$ is an element of $\mathcal{T} = \left( \frac{\mathfrak{t}_{\mathbb{C}}^\vee}{2\pi i \rho(\mathfrak{g})} \right)^{\mathcal{W}(T, G)}$ (this is called the **FI-theta parameter** in the physics literature) For $\mu : V \to \mathfrak{g}^\vee$ the moment map and $\zeta \in \text{Re}(\mathcal{T})$ a regular value, the **Higgs branch** is a certain intersection $X_\zeta$ of the preimage $\mu^{-1}(\zeta)$ inside $V$ (what is it?). The **Coulomb branch** is the union of loci in $\mathcal{T}$ where the GLSM is singular (I think). Set $\widetilde{W}_{\text{eff}}(\sigma, t) = -t(\sigma) + 2\pi i \rho_{W}(\sigma)\cdot\sum_{Q_{j} \in \mathfrak{h}^\vee} Q_{j}(\sigma)(\text{log}(Q_{j}(\sigma) - 1))$, where $t$ is as above and $\sigma \in \mathsf{Z}(\mathfrak{g}_{\mathbb{C}})$. Then the **stringy Kähler moduli space** is $\mathcal{M}_{K} = \mathcal{T} \setminus \Delta$, where $\Delta$ is the union of critical loci for $\widetilde{W}_{\text{eff}}^H$ over all subgroups $H$ of $G$. Remark: The real locus of the stringy moduli space has finitely many connected components $P_i$ called **phases**. By the relation between symplectic and GIT quotients, each phase is a possible linearisation for the GIT quotient. The imaginary locus has smoothness properties allowing one to interpolate through the real phases by passing through the imaginary locus. #### GLSM $\to$ D-branes (not used later) A **brane** is a 4-tuple $B = (M, Q, \rho, r_*)$, where - $M$ is a $\mathbb{Z} / 2\mathbb{Z}$-graded free $S$-module ($S$ as above) - $Q \in \text{End}^1_S(M)$ (this subscript usually indicates a distinguished subgroup of the endomorphism group) - $\rho_M : G \to GL(M)$ is a complex representation - $r_{*} : U(1) \to GL(M)$ is a complex representation such that $Q^2 = W$, $Q$ is $G$-equivariant, etc. - There are limiting points $p_i$ attached to each phase above, which have associated triangulate categories $D_i$ (these are really associated to the Higgs branch of regular values in the phase). There are various conjectural equivalences of these $D_i$ to triangulated categories of algebraic invariants, depending on the branch. - Conjecture: There are essential surjections $\pi_i : D \to D_i$ (from total triangulated category to triangulated category of the phases) and natural equivalences $D_{i} \xrightarrow{\sim} D_{j}$. - Define the **hemisphere partition function** $$ \begin{aligned} Z_{D^2} &: K_{0}(D) \times \mathcal{M} \to \mathbb{C}, \\ Z_{D^2}([B], t) = &\int_{\gamma \subseteq \mathfrak{t}_{\mathbb{C}}}d^r\sigma \prod_{\alpha \in \prod^+} \alpha(\sigma)\sinh(\pi \alpha(\sigma)) \prod_{j=1}^m \Gamma\left( i Q_{j}(\sigma) + \frac{R_{j}}{2} \right)\exp(it \sigma) f_{B}(\sigma) \end{aligned} $$ ( not sure what all of the parts here are). Conjecture: To each limiting point $p_i$, there should be an associated cohomological field theory: - a state space $H_i = HH_*(D_i)$ (Hochschild homology) with a pairing $\langle \ , \ \rangle: H_{i} \times H_{i} \to \mathbb{C}$ - for all $i$, $Z_{D^2}([B], t) = \langle \text{ch}(\pi_{i}(B)), \hat{\Gamma}_{i} \circ J_{i}(t) \rangle$, where $\text{ch}$ is the Chern character, the $J_i \in H_i[t][[e^t]]$ are certain *J-functions* and $t \in P_i + (i \mathbb{R}/ \mathbb{Z})^{\text{dim}(\mathcal{M}_{K})}$. #### Elliptic Fibrations - In the following examples, we will use an **elliptic normal curve**. For our purposes, this is a smooth projective curve $E$ of genus 1 and degree $N \geq 3$, so that $E$ can be immersed in $\mathbb{P}^{N-1}$ with image not contained in a hyperplane. - Example: If $N = 3$, $E$ is a cubic in $\mathbb{P}^2$. #### Pfaffian varieties Let $A$ be a $(2r+1)$-dimensional square matrix, skew-symmetric with entries in $(\mathsf{Sym}^\bullet(x_0,\dots, x_{r}))_{1}$ (degree 1 polynomials in $r+1$ variables). Let $M_i$ be the $2r\times 2r$ minor obtained by deleting the $i$th row and column from $A$, $\text{pf}(M_{i})$ its **Pfaffian** (choice of square-root for $\text{det}(M_{i})$). We define the **Pfaffian variety** $D_{r}(A)$ of $A$ to be $V(\prod_{i=1}^{2r+1}\text{pf}(M_{i}))$, the vanishing locus of all of the Pfaffians. #### Homological projective duality for elliptic normal curves in degree Let $V$ be a 5-dimensional complex vector space, $L \subseteq \bigwedge^2 V^\vee$ a 5-dimensional subspace, $L^\perp$ its orthogonal complement. Then we have a diamond

of inclusions with $X_0$ our elliptically fibred surface (I think ), the Plücker embedding from the Grassmannian and the induced map of projective spaces from the inclusion $L^\perp \subseteq \bigwedge^2 V$. The dual picture is the diamond

and we find that $\text{Pf}(2, V^\vee)$ is dual to the Grassmannian, $\mathsf{D}^{\flat}(\text{Coh}(X_{0})) \cong \mathsf{D}^{\flat}(\text{Coh}(Y_{0})).$ In the GLSM version of this homological duality setup (see [Hori Knapp '13](https://arxiv.org/pdf/1308.6265)), $G = U(2)$ with stringy Kähler space $\mathcal{M}_{K} = \mathbb{C}^* / e^{-t}$, $e^{-t} = \frac{1}{2}(11 \pm 5\sqrt{5})$. The Higgs branch is $Y_0$. ### General Elliptic Fibration Picture The correct setting for an elliptic fibration is a flat proper holomorphic surjection $\pi: X \twoheadrightarrow B$. There is a minimal integer $N > 0$ such that a divisor $D \subseteq X$ exists with $\pi|_{D} : D \xrightarrow{N : 1} B$ an $N$-fold cover. If $N = 1$, we call $\pi$ an **elliptic fibration**. Equivalently, there is a line bundle $L \to X$ with $\text{deg}(L|_{C}) = N$ for every fibre $C$ of $\pi$, $D\cdot C = N$ (has intersection multiplicity $N$). Intersection theory characterisation of Calabi-Yau elliptic fibrations ([Oguiso '93](https://archive.mpim-bonn.mpg.de/id/eprint/382/1/preprint_1993_72.pdf)): Let $X$ be a smooth projective Calabi-Yau threefold, $D \geq 0$ a divisor on $X$ satisfying $D^3 = 0$, $D^2 \not \equiv 0$, $D\cdot c_{2} > 0$. Then there is a base space realising $X$ as an elliptic fibration. - People are interested in a physics interpretation of these examples for the modular bootstrap and for studying the topological string partition function - Knapp, Schimanneck and Scheidegger focus especially on the $N = 5$ elliptic normal curve because the Calabi-Yau cannot be realised as a singular toric variety (most known mirrors can) - They constructed 13 pairs of such spaces and computed their Hodge numbers and other diffeomorphism invariants Theorem([Knapp, Schimannek, Scheidegger '21](https://arxiv.org/pdf/2107.05647)): Let $\pi : X \to B$ a smooth projective Calabi-Yau threefold which is an elliptic fibration admitting a 5-section ( this should be equivalent to the fibres being degree 5 elliptic normal curves). Then: - $X$ is a determinantal subvariety of a Grassmannian bundle - $X$, $\mathcal{M}_{K}$ can be studied using a $U(M)\times U(1)^n$ GLSM ($M$ a certain $D$-brane) - these fibrations come in pairs $(X, Y)$, which are **conjecturally mirror duals** - there is also a statement about Fourier-Mukai transforms I missed ### Open Questions ( this summary should be close to correct but I'm kind of guessing what the notation means) In general, we so far usually have a picture as below: We have a mirror space $Y$ and an analytic non-Kähler resolution $\overline{Y}$, but we can get an associated algebro-geometric space by taking the noncommutative space associated to a nonzero element $\alpha \in \text{Br}(Y)$ which will have $\mathsf{D}^{\flat}(\text{Coh}(\overline{Y})) \cong \mathsf{D}^{\flat}(\text{Coh}(Y_{\text{nc}, \alpha}))$. Then we can pass to a smooth deformation $Y'$ of $\overline{Y}$. The correct and precise version of this is in [Thomas Calabrese '16](https://arxiv.org/pdf/1408.4063).

The things we want to know are: - What are the bounded derived categories of singular elliptic fibrations? - Is it possible to calculate topological invariants using only the data of the $D$-brane? - Can we work out $\text{Br}(Y)$ and the noncommutative scheme structure from the GLSM? - Can we work out the Fourier-Mukai transform using matrix factorisation in the GLSM?

Deformation Functor under Completion

Mon, 16 Feb 2026 00:00:00 GMT

Today I would like to discuss the proof that if $R$ is a local ring with residue field $k$, the functors $h_R$ and $h_{\hat{R}}$ are isomorphic, where $h_S : \mathsf{Art}/k \to \mathsf{Set}, h_{S}(A) = \text{Hom}_{k}(S, A)$ is the functor sending an artinian local ring with residue field $k$ to the set of maps into it from $S$. We will see that this boils down to the fact that artinian local rings are complete (I will not prove it this way, but trust that readers can see the underlying idea), but I would also like to talk about identifying this as the correct deformation functor. On the one hand, $h_{S}$ is an obvious choice since it's the restriction of the contravariant representing functor for $\text{Spec} \ S$ to $\mathsf{Art}/k$, which we use because the category of schemes has fibre products. On the other, the completion $\hat{R}$ is universally mapped *into*, so the functor $h^{S}(A) = \text{Hom}_{k}(A, S)$ might seem more appropriate. Let's see what happens if we expand the universal property. Working with our local ring $R$, for $A$ Artin we have $$h^{\hat{R}}(A) = \text{Hom}_{k}(A, \hat{R}) = \text{Hom}_{k}(A, \varprojlim_{n} \ R/\mathfrak{m}_{R}^n) = \varprojlim_{n} \ \text{Hom}_{k}(A, R/\mathfrak{m}_{R}^n),$$ but nilpotence of the maximal ideal in $A$ does not allow us to relate back to $R$ since it does not furnish us with a choice of element in $R$ corresponding to the image of an element in $A$ (reason: two elements of $R$ can correspond to the same sequence in $\hat{R}$; their difference lives in $\cap_{n \geq 0}\mathfrak{m}_{R}^n$, so when this ideal is nonzero we have no preferred choice of class representative). So we see that our proposition is unlikely to hold if we use the $h^R$ functors without assuming $R$ itself is Artin (this would make $\cap_{n \geq 0}\mathfrak{m}_{R}^n = 0$). Local rings with fixed residue field arise naturally in geometry but artinian rings appear in more specialised contexts, so we see that the contravariant hom-functor-functor is not ideal for deformations. Back to the exercise, we show that the completion map $\varphi_{R}: R \to \hat{R}, r \mapsto (r \pmod{\mathfrak{m}_{R}}, r \pmod{\mathfrak{m}_{R}^2}, \dots)$ induces our isomorphism of functors. It suffices to show that $$ \varphi_{R}^* : h_{\hat{R}}(A) \to h_{R}(A) $$ is an isomorphism for every Artin local ring $A$ with residue field $k$. Fix $n$ minimal so that $\mathfrak{m}_{A}^n = 0$. For distinct maps $f, f': \hat{R} \to A$, there is a sequence $\tilde{r} = (r_{0}, r_{1}, \dots) \in \hat{R}$ such that $f(\tilde{r}) \neq f'(\tilde{r})$, and we can construct an element $s$ of $R$ with $f\varphi_{R}(s) \neq f'\varphi_{R}(s)$ as follows. Choose any coset representative $s_{0}$ of $r_0$, so that $\tilde{r} - \varphi_{R}(s_{0}) \in \mathfrak{m}_{R}\hat{R}$. Then if $s_1$ is a coset representative for $r_1 - [s_0]$ in $R/\mathfrak{m}_{R}^2$ we have $s_1 \in \mathfrak{m}_{R}$, so that $$ \begin{aligned}\varphi_{R}(s_0 + s_1) &= (r_{0}, r_1 - s_{0} + s_{0}, \ s_{0} + s_{1} \pmod{\mathfrak{m}_{R}^3}, \dots) \\ &= (r_{0}, r_{1}, s_{0} + s_{1}, \dots).\end{aligned} $$ We can continue this process to approximate $\tilde{r}$ to any finite number of entries, so let us take $s$ with $\varphi_R(s) = (r_0, \dots, r_n, s \pmod{\mathfrak{m}_{R}^{n+1}}, \dots)$ in $\hat{R}$. We claim that this $s$ satisfies $f\varphi_{R}(s) \neq f'\varphi_{R}(s)$. This follows as soon as we know that for any $f : \hat{R} \to A$, $f(\mathfrak{m}_{R}^n) \subseteq \mathfrak{m}_{A}^n$ because then $f(\tilde{r})$ is independent of the entries of $\tilde{r}$ past $n$. But this statement is clear, for $f(\mathfrak{m}_{R}) \subseteq \mathfrak{m}_{A}$ and multiplying gives the same inclusion for general $n$. Thus $\varphi_{R}^*$ is injective. Now consider a general map $g: R \to A$. By what we have just mentioned, $g(\mathfrak{m}_{R}^n) = 0$, so the idea is to define a map $\tilde{g}: \hat{R} \to A$ which only uses information from the first $n$ entries of the sequences in $\hat{R}$. Explicitly, for $\tilde{r} \in \hat{R}$ we set $\tilde{g}(\tilde{r}) = g(r)$, where $r$ is any element of $R$ for which $\varphi_{R}(r)$ agrees with $\tilde{r}$ to $n$ places. That $g(\mathfrak{m}_{R}^n) = 0$ ensures this is a well-defined map, and additivity, the multiplicative property, and the $k$-homomorphism condition for $\tilde{g}$ all follow from these properties for $R$ and from the homomorphism $\varphi_{R}$. Since $\tilde{g}\varphi_{R} = g$ by construction, we have the surjectivity of $\varphi_{R}^*$.

Hartshorne Suggestions Feb26

Fri, 13 Feb 2026 00:00:00 GMT

import * as images from "../../images/hartshorne" Old exercise suggestions from Hartshorne for practice and possibly future blog posts. {/* I really need to use eg GraphQL to do this programmatically*/} ### Chapter I, section 3:
### Chapter II, setting up $sections 1-4$:
### Chapter II, calculations and stronger invariants $sections 5-8$:
### Chapter III, developing cohomology $sections 1-4$:
### Chapter III, results from cohomology $sections 5-9$:

Notes on Thesis Feedback

Thu, 12 Feb 2026 00:00:00 GMT

*This is a page of old notes I collected on feedback from my master\'s thesis, which I put together shortly after taking the time to understand the comments. I later corrected the mistakes and my understanding of the feedback in late June 2026, so the below notes are not current. I especially like having these old notes because it shows how much your understanding can change in a short time!* (Definition of algebraic group): Note that I am defining an algebraic group to be a finite type $k$-scheme which is also a group scheme. My remark about the functor of points factoring through groups should then be interpreted as follows: if $G$ is a finite type $k$-scheme, it is an algebraic group if its functor of points factors through $\mathsf{Grp}$. This is not as explicit as it should be in the submitted thesis. (Example 3.12): The strong reaction to this section for such a strong error makes sense, but I have to disagree that the exposition is problematic. The exposition seeks to accomplish two things: 1. Mention that there is a complete classification of reductive groups in the generality of a local field (main problem with this: I don't define or use local fields, I just want a correct statement) 2. Conclude that $GL_{n}(k)$ and $SL_{n}(k)$ are reductive, as these are the groups we use later. The reasoning for 2 used to get that $GL_{n}(k)$ is reductive is incorrect and the subsequent argument is nonsense (though not hopeless). Having a faithful representation in $GL_n(k)$, one then needs to analyse when a subgroup of $GL_n(k)$ has trivial unipotent radical. Unfortunately, at the time I didn't properly understand how one works out the radical or the unipotent radical of a linear algebraic group: I read a lot of sources, but couldn't find a single place where an example was simply and cleanly laid out. I should have worked on this earlier, but figuring it out last minute and being unable to find a reference didn't work out. $\textcolor{#AD42DB}{\text{Homework: Learn how to really show }GL_n, SL_n \text{ are reductive}}$. (Lemma 4.14): The feedback is correct, the exact sequence used need not be split. This isn't an easy exercise in algebra; both Zhu's notes and Timm Peerenboom's thesis break the proof into multiple steps with detailed discussion. I should have been more careful and not let Beauville and Laszlo's wording get to me. There are two parts to show: 1. If $W$ is finitely generated, being a lattice in the sense that $\bigcup_{n \in \mathbb{Z}}z^{n}W = R(\mkern-3mu (z)\mkern-3mu)^{\oplus r}$ is equivalent to having $z^nR[\mkern-3mu [z]\mkern-3mu]^{\oplus r} \subseteq W \subseteq z^{-n}R[\mkern-3mu [z]\mkern-3mu]^{\oplus r}$ for some $n > 0$. 2. $W$ is finite projective over $R[\mkern-3mu [z]\mkern-3mu]$ if and only if $Q = z^{-n}R[\mkern-3mu [z]\mkern-3mu]^{\oplus r}/W$ is a projective $R$-module. The two notions of projectivity in (2) was a subtlety I didn't really appreciate when putting together the thesis. The proof itself is also more difficult, so we'll start with (1). 1. ($\longrightarrow$): Let $e_i$ be the $i$th standard basis vector for $R(\mkern-3mu (z)\mkern-3mu)^{\oplus r}$, and choose $n_i$ so that $e_i \in z^{n_i}W$. If $n_i \geq 0$ then we have $z^{n_i}W \subseteq W$ and $e_i$ is in $W$; if we let $\mathcal{N}$ be the set of all $n_i$ which are negative, then $N = \text{min}_{n \in \mathcal{N}} \ n$ is such that $R[\mkern-3mu [z]\mkern-3mu]^{\oplus r} \subseteq z^N W$ for $z^iW \subseteq z^jW$ when $j < i < 0$. Multiplying by $z^{-N}$ gives $z^{-N}R[\mkern-3mu [z]\mkern-3mu]^{\oplus r} \subseteq W$. For the other inclusion, since $W$ is finite rank it admits an $R[\mkern-3mu [z]\mkern-3mu]$-basis of vectors $\alpha_{1}, \dots, \alpha_{s}$ written in the standard coordinates for $R(\mkern-3mu (z)\mkern-3mu)^{\oplus r}$, which have a total of $rs$ Laurent series components. Since each Laurent series has a finite negative tail, we can choose $m \in \mathbb{Z}$ so that $z^m$ has the least negative exponent in any of the Laurent series. It follows that $W \subseteq z^mR[\mkern-3mu [z]\mkern-3mu]^{\oplus r}$, and by choosing $M = \text{max}\{m, -N\}$ we can make the $z$-exponents in our inclusions symmetric. ($\longleftarrow$): Multiplying the inclusions by $z^{-n}$ gives $R[\mkern-3mu [z]\mkern-3mu]^{\oplus r} \subseteq z^{-n}W$, and from this point we can clearly multiply by any $z^{-m}$, $m \in \mathbb{N}$ on both sides to show that $\bigcup_{n \in \mathbb{Z}} z^{-n}W = R(\mkern-3mu (z)\mkern-3mu)^{\oplus r}$. Since $W \subseteq z^{-n}R[\mkern-3mu [ z]\mkern-3mu]^{\oplus r}$ which is a free $R[\mkern-3mu [z]\mkern-3mu]$-module of rank $r$ (generated by the $z^{-n}e_i$), we know that $W$ is finitely generated by $\leq r$ generators, as $R[\mkern-3mu [z]\mkern-3mu]$ is a PID. (this works on $k$-points, but I don't think it's true in general. I've kept this here as a reminder of this fact). 2. \[I didn't finish this part at the time.\] (Lemma 4.18): A preceding comment is incorrect, permutation matrices have determinant $\pm 1$ so not all of them are in $SL_r$. Regardless, the Weyl group of $SL_r$ is $S_n$ which could be used to justify this line of reasoning. Better would be to omit this comment since it seemingly involves more reasoning than I expected. The comment provided is confusing to me as we **want** $d_2 - a < d_3$, but the discussion in my proof is not so clear. We would only have a problem if $d_{1}' \geq d_{2} -a \implies d_{1}' + a = 2d_{1}' - d_1 \geq d_{2}$ and there is no reason this couldn't happen.

Total Space of Bundles on Proj(S)

Sat, 31 Jan 2026 00:00:00 GMT

I have a lot of trouble remembering how the Proj functor works and how to think about sheaves of $\mathcal{O}_X$-modules on these spaces. Part of what makes this hard is that everybody finds it technical and explains it in their own way; in this document, I'll write things down in *my* own way (lifted from Hartshorne but more explicit, probably) so that I have a reference.
Let $S$ be a graded ring, and let $\mathsf{Proj}(S)$ be the set of homogeneous primes in $S$. To topologise $\mathsf{Proj}$ we take the sets $V(\mathfrak{a})$ as our closed sets, and to get a structure sheaf we define $$ \mathcal{O}(U) = \bigl\{s : U \to \coprod_{\mathfrak{p} \in U}S_{(\mathfrak{p})}, \substack{s(\mathfrak{p}) \ \in \ S_{(\mathfrak{p})} \text{ for all } \mathfrak{p} \in \mathsf{Proj}(S) \\ \text{locally } s \text{ is a quotient } a/f \\ \text{of homog elements in } S \text{ of same degree}}\bigr \}, $$ where $S_{(\mathfrak{p})}$ is the ring of degree 0 elements in $T^{-1}S$ and where $T$ is the multiplicative system of homog elements in $S \setminus \mathfrak{p}$. One then shows that this makes $\mathsf{Proj}(S)$ a locally ringed space, having distinguished open sets $D_+(f) = \{\mathfrak{p} \ | \ f \not \in \mathfrak{p} \}$ for homogeneous $f$ such that $D_+(f) \cong \text{Spec} \ S_{(f)}$. Sheaves of modules $\tilde{M}$ on $\mathsf{Proj}$ are described in the same way as the structure sheaf $\mathcal{O}$: for a graded $S$-module $M$, we define $M_{\mathfrak{p}}$ in the analogous way and look at stalk-functions $m : U \to \coprod_{\mathfrak{p} \in U} M_{(\mathfrak{p})}$. A feature that distinguishes $\mathcal{O}$-modules from $\mathcal{O}_X$-modules on $X$ affine is existence of **Serre twists**. For $n \in \mathbb{Z}$, we can twist the grading of $S$ by making the $m$th component of $S$ into the $m+n$th component of $S(n)$. This twisting is compatible with localisation, so it extends to give twists of $\mathcal{O}$. Finally, we get twists of $\mathcal{O}$-modules by setting $\tilde{M}(n) = \tilde{M} \otimes \mathcal{O}(n)$. Structurally, $\tilde{M}$ and $\tilde{M}(n)$ are quite different sheaves: the former has local and global values determined by the degree 0 parts of $M$ and its localisations, while the latter has local and global values from the degree $n$ parts.

Research Suggestions Jan26

Tue, 27 Jan 2026 00:00:00 GMT

Priority High:

- Gromov-Witten theory: Understand some computations and the virtual fundamental class.
Resources: * Cox Katz – _Mirror Symmetry in Algebraic Geometry_ * Fulton Pandharipande – _Notes on Stable Maps and Quantum Cohomology_ * Kontsevich Manin – _Gromov-Witten Classes, quantum cohomology, and enumerative geometry_ - The set-up of a symplectic resolution controls deformation theory $should give a finite-dimensional moduli space$... why is this important for setting up quantum cohomology? - How do we think about and write down connections in the diff geo setting first, but also in the alg geo setting? - Learn some basics of equivariant K-theory and Borel-Moore homology as needed. - Learn where the geometry of the Slodowy slice comes from and some examples. - Work out quantum cohomology in the $$T^*\mathbb{P}^1$$ example. - Read about Steinberg variety in Chriss-Ginzburg. - Learn about $tangent-$obstruction theories, understand the motivating example.

Priority Medium:

- Why do $SL_2$-subgroups lead to embedded rational curves? - How does $G$-action equivariance rigidify deformations, why are Hilbert schemes examples of equivariant symplectic resolution? - What are correspondences and the correspondence algebra? Explicit examples please. - Understand the connection between monodromy and derived equivalences (*Horja - Mirror Symmetry, Hypergeometric Functions* (keywords, not actual title)) - Read Kaledin – *Symplectic singularities from the Poisson point of view* to understand Poisson variety structure on base space of the resolution. - Look at Maulik-Okounkov framed sheaves, Maulik et al on GW/DT Correspondence for Toric 3-Folds to understand decomposition of the shift operators. - Learn about Springer sheaf and IC decomposition.

Priority Low:

- Why are symplectic deformations classified by $H^2_{dR}(X)$? - Learn something about mirror symmetry (Cox-Katz, Vakil Zaslow et al – *Mirror Symmetry*, Gross – *Calabi-Yau Manifolds and Mirror Symmetry*) - Read about quantum cohomology of flag varieties (ideally a similar story) - Look at Negut – *Laumon Spaces and the Calogero-Sutherland Integrable System* - Kodaira-Spencer map and examples

Tangent-Obstruction Theories

Wed, 21 Jan 2026 00:00:00 GMT

If $R$ is a local $k$-algebra with residue field $k$ and finite tangent space $(\mathfrak{m}_{R}/\mathfrak{m}_{R}^2)^\vee$ of dimension $d$, then $R$ is a quotient of a power series ring $S = k[\mkern-3mu [t_{1}, \dots, t_{d}]\mkern-3mu]$ by an ideal $J \subseteq (t_{1}, \dots, t_{d})^2$. Let $T$ be the tangent space of $S$ and write $n = (t_1, \dots, t_{n})$. Then we have the following
**Theorem**: For every small extension $$0 \to M \to B \to A \to 0$$ (ie a surjection of Artinian $k$-algebras with kernel $M$ annihilated by $\mathfrak{m}_{B}$), there is a short exact sequence $$0 \to T\otimes_{k} M \to \text{Hom}_{k}(R, B) \to \text{Hom}_{k}(R,A) \xrightarrow{ob} (J/nJ)^\vee \otimes_{k} M,$$ which is functorial in small extensions. What this theorem says is that: - In this setting, a map $\varphi : R \to A$ lifts to $B$ if and only if $\text{ob}(\varphi) = 0$ - When a lift of $\varphi: R \to A$ exists, the set of all lifts to $\varphi$ has a transitive action by $T\otimes_{k}M$, an is thus an affine space ($T\otimes_{k}M$ is a $k$-vector space, so the set of lifts over $\text{Hom}_{k}(R, A)$ is a fibration with affine fibres) The proof of the theorem has three ingredients: for the first we use that there are no relations among the $t_i$ in $S$ to lift maps $S \to A$ to $B$. For the second, we show that the difference between two lifts of the same map is a derivation $S \to M,$ and the tangent space appears as the kernel because $\text{Der}_{k}(S,M) \cong T_{S}\otimes_{k} M$. Finally, we notice that all lifts agree on $J$ with $nJ$ in the kernel of every lift by smallness, so the common values of a lift for $\varphi$ give an element of $\text{Hom}_{k}(J/nJ, M) \cong (J/nJ)^\vee \otimes_{k} M$. If we plug in the local ring at a point $R = \mathcal{O}_{X, \ p}$ of a smooth variety of dimension $d$ (or the ring at a smooth point of a general variety), then taking $A = k$ and $B = k[\varepsilon]/\varepsilon^2$, $\text{Hom}_{k}(R, A) = \text{pt}$ and $\text{Hom}_{k}(R,B)$ is the tangent vectors to $X$ at $p$ (HS II.2.8). Since $T \otimes_{k} M = T \otimes (e) \cong T$ as vector spaces, we have $T \cong \text{Hom}_{k}(R, B)$ and the obstruction map vanishes. This says that all tangent vectors in $\text{Hom}_{k}(R, B)$ sit over the same point $p$ (which we already knew). But notice that we can also use the tangent-obstruction sequence to study lifts to higher-order tangents/jets. To proceed, we need to take on faith that this theorem reflects expected properties about deformations: there should be certain 'directions' we can deform in and others for which there are obstructions, and we should have a multidimensional 'space' of deformations. Perhaps we know this by understanding that deformations correspond to classes in $H^1(X, T_{X})$. We should be completely satisfied with this theorem and look for other situations where it holds, leading us to define: A deformation functor $D$ has a **tangent-obstruction theory** if there are finite-dimensional $k$-vector spaces $T_1$ (the **tangent space**) and $T_2$ (the **obstruction space**) so that: 1) For all small extensions $$0 \to M \to A \to B \to 0,$$ there is a corresponding exact sequence of sets $$T_{1} \otimes_{k} M \to D(B) \to D(A) \xrightarrow{ob} T_{2}\otimes_{k}M.$$ 2) If $A = k$, the map $T_{1} \otimes_{k}M \to D(B)$ is injective. 3) The tangent-obstruction sequences are functorial in small extensions. Sometimes we will be able to develop a tangent-obstruction theory where $T_1$ and $T_2$ need not be finite-dimensional. In this case, we call the theory a **generalised tangent-obstruction theory** (our tangent-obstruction theorem is *more general* than the motivating case).

Hall Algebras and Rational Points of Grassmannians

Sat, 13 Dec 2025 00:00:00 GMT

This week I attended AustMS, but I still had some time on the long tram trips to do a bit of study. I mostly spent this time learning what a Hall algebra is and studying the example of nilpotent representations for the Jordan quiver. Thus far, my best summary would be that the *Hall algebra* of a suitable abelian category is an associative algebra which contains information about the short exact sequence structure of the category. Let $$\mathcal{A}$$ be an abelian category. To make the objects of $$\mathcal{A}$$ into an associative $$k$$-algebra tracking short exact sequences, one natural idea is to take the free $$k$$-vector space $$\mathbb{H}_{\mathcal{A}} = \bigoplus_{M \in \textnormal{Ob}(\mathcal{A})}kM$$ with basis the objects of $$\mathcal{A}$$, and to define the product on basis elements by $$ M*N = \sum\limits_{\text{extensions } K \text{ of } M \text{ by } N} a^K_{MN}K \ \text{ for constants } a^K_{MN} \in k.$$ We use the direct sum to ensure that calculating products and sums requires determining finitely many entries. Our goals for this definition are that: - The choices of structure constants $$a^K_{MN}$$ make the multiplication associative, and - The structure constants record properties of the short exact sequences/extensions in $$\mathcal{A}$$. With this in mind, letting $$a^K_{MN} = \#\{ \phi \in \text{Ext}(M,N) \ | \ \phi \text{ has middle term K } \}$$ allows the coefficients to enumerate extensions.

Introductory Representation Theory

Sat, 01 Nov 2025 00:00:00 GMT

Let $G$ be a group. A **representation** of $G$ is a homomorphism

$$\rho: G \to GL_n(V),$$

where $V$ is a (finite dimensional) complex vector space. Examples: For any group $G$, we have the: - trivial representation $G \to GL_1(\mathbb{C}), \ g \mapsto 1$ - regular representation $r_G : G \to GL_{|G|}(\mathbb{C}G), \ r_g(g)(e_h) = e_{gh}$ - permutation representations: given a group action $G \curvearrowright X$, $p : G \to GL_{|X|}(\mathbb{C}X), \ p(g)(e_x) = e_{g\cdot x}$. In scope I would also like this to include a discussion of character theory.

Induced Representations, Mackey Theory

Sat, 01 Nov 2025 00:00:00 GMT

Where I'll write about Mackey theory and generalisations.

Schur-Weyl Duality, Tableaux, GLn

Sat, 01 Nov 2025 00:00:00 GMT

I set up this file to help with exam revision, but didn't end up using note-taking to prepare. I'm hoping to refine this once I've learnt the Okounkov-Vershik approach!!