TAGS 2024

Scattered notes taken during the talks at the Texas Algebraic Geometry Symposium (TAGS) 2024

Talk 1: David Eisenbud
1. Infinite Resolutions
Talk 2: Chiara Damiolini, A gentle introduction to conformal blocks
1. Classical picture of conformal blocks
2. Generalizations
Talk 3: Giorgio Ottaviani, The Hessian Map
1. Consequences of Gordan-Noether
Talk 4: Local heights on hyperelliptic curves for quadratic Chabauty
Talk 5: KSBA moduli spaces of stable varieties
Talk 6: Hannah Larson, Cohomology of Moduli Spaces of Curves
Talk 7: Alicia Lamarche. Wonderful compactifications, toric varieties and derived categories
Talk 8: David Eisenbud. Infinite Resolutions.

Talk 1: David Eisenbud

Take \(R\) to be a commutative ring, set \(S = S_n = k[x_1,...,x_n]\) and let \(M\) be a finitely generated \(R\)-module. We'll be looking at free resolutions of \(M\):

\[\begin{aligned} ... \to F_1 \to F_0 \to M \to 0 \end{aligned}\]

where \(F_i\) are free finite rank modules. If \(R = S_n\), then there always exists a free resolution with \(F_{n+1} = 0\).

Theorem: (Buchsbaum, Eisenbud). Suppose you have a complex

\[\begin{aligned} 0\to F_n\xrightarrow{\varphi_n}...\xrightarrow{\varphi_0} F_0 \end{aligned}\]

of finitely generated \(R\)-modules. This is a resolution if and only if the following hold:

\(\operatorname{rank} F_i = \operatorname{rank} \varphi_{i+1}+\operatorname{rank} \varphi_{i}\)
\(I_{\operatorname{rank} \varphi_i}(\varphi_i)\) has depth \(\geq i\), \(\{x\in \MaxSpec R \mid \operatorname{rank} \varphi_i(x) < \operatorname{rank} \varphi_i\}\) has codimension \(\geq i\).

Theorem: (Hilbert-Burch). A If \(0\to R^{n+1}\xrightarrow{\varphi_2} R^n \xrightarrow{\varphi} R \to R/I\to 0\) is a free resolution then there exists a nonzero divisor \(a\) such that \(I = a\cdot I_{n-1}(\varphi_2)\) and conversely if \(\operatorname{depth}I_{n-1}(\varphi_2) \geq 2\).

This should be viewed as some sort of "structure theorem" for free resolutions of a quotient of a local or graded ring.

Infinite Resolutions

Let \(R = k[x]/x^3\) be a polynomial ring in one variable. Then

\[\begin{aligned} ...\xrightarrow{x\cdot } R\xrightarrow{x^3\cdot} R\xrightarrow{x\cdot}R\to R/x \cong k \to 0 \end{aligned}\]

is an infinite resolution.

Theorem: Let \(R = S/f\) where \(S\) is a regular local ring, \(f\in S\) is any element vanishing to order \(\geq 2\). Then \(f\cdot Id = AB\) for some square matrices \(A\) and \(B\) and the free resolution of \(\operatorname{coker} A\) is ...

Difficult to see from the back of the room

Talk 2: Chiara Damiolini, A gentle introduction to conformal blocks

A story: two mathematicians. They hike. In forest. There's a lion. They run away (as you would). Lion is faster though. One mathematician has an idea, runs back to the lion, whispers in its ear. Lion stops, turns around and runs away. The mathematician whispered something about dinner.

The name "conformal blocks" is bad, says Chiara. We should call them "coinvariants" instead. Everything in this talk is joint with A. Gibney, D. Krashan and N. Tarama.

Classical picture of conformal blocks

There are two inputs, geometry and representation theory.

Geometric input: a smooth projective curve \(C\) over \(\mathbb C\), i.e. \(C\in \mathcal M_{g,n}\).
Rep theory input: \(\mathfrak g\) a simple Lie algebra, \(\ell \in \mathbb Z_{> 1}\), \(W_{\lambda_1}, ..., W_{\lambda_n}\) irreducible representations of \(\mathfrak g\) all of weight \(\leq \ell\). Our typical example will be \(\mathfrak{sl}_2\), the traceless \(2\times 2\) matrices over \(\mathbb C\). Always think of a representation of \(\mathfrak{g}\) as a vector space with a \(\mathfrak g\) action.

("Level \(\ell\)" means that nilpotents must act by zero after applying them \(\ell + 1\) times.)

We can pair these two datum together to produce a vector space.

Given a curve with \(n\)-marked points \((C, p_1,...,p_n)\), we pair each marked point with one of the irreducible representations \(W_{\lambda_i}\). This is called a \(\mathfrak g\)-decorated curve. This produces a vector space of coinvariants \(\mathbb V(C, P, \mathfrak g, \ell, W)\). This vector space is called the vector space of coinvariants. The space of conformal blocks is the dual \(\operatorname{Hom}_{\mathbb C}(\mathbb V, \mathbb C)\).

Some fun facts.

(Theorem:) \(\mathbb V\) defines a quasi coherent sheaf on \(\overline{M}_{g,n}\).
It's actually coherent, not merely quasi-coherent.
There exists a projective flat connection on \(\mathbb V|_{M_{g,n}}\).
Conditions (2) and (3) imply that \(\mathbb V\) is a vector bundle on \(\mathcal M_{g,n}\). (NOTE \(\mathcal M_{g,n}\) not \(\overline{\mathcal M}_{g,n}\).)
There is something called Factorization which tells us that to compute the rank of \(\mathbb V\) as a bundle over \(\mathcal M_{g,n}\), it suffices to understand conformal blocks over \(\overline{\mathcal M}_{0,3}\): \(\mathbb V(\mathbb P^1, 0,1,\infty, \mathfrak g, \ell, A,B,C)\).
The space of conformal blocks over a curve \(C\) with a single marked point \(P\), \(\mathbb V(C,P,\mathfrak g, \ell, \mathbb C)^*\), is canonically isomorphic to \(H^0(\mathsf{Bun}_{G,C}, \mathcal L_\ell)\) where \(G\) is the simply connected Lie group associated to \(\mathfrak g\).
On \(\overline{\mathcal M}_{g,n}\), \(\mathbb V\) are globally generated

Note: The fact (2) is not obvious. There is a thickening procedure during which the points \(P_i\) are replaced with formal disks \(\mathbb C[[t]]\), together with corresponding modifications to the lie algebra \(\mathfrak g\) and the representations \(W_i\), so that you end up with a finite dimensional thing is interesting.

Generalizations

The first generalization might be to go from \(G\) to a parahoric group \(\mathcal G\). This is a class of groups which still have the uniformatization theorem, a key results in proving the isomorphism in (6) above.

Another generalization asks: does it make sense to look at conformal blocks over cuspoidal curves?

In fact, there is some work done by Tarasca which replaces curves with abelian varieties – beyond curves conformal blocks.

The last generalization Chiara mentioned replaces the representation theory data with rep theory data from vertex operator algebras. This makes sense because vertex operator algebras are naturally objects which live over disks. In this generalization, you still get a quasi-coherent sheaf over \(\overline{M}_{g,n}\), although it's quite hard to handle the case of nodal curves in this setting. If you impose the condition that your vertex operator algebras are "\(C_2\) cof., then you retain coherentness too, and there always exists a projective flat connection. Factorization still works. The current big question is to "go beyond rationality".

Talk 3: Giorgio Ottaviani, The Hessian Map

Given a polynomial \(f(x_0,...,x_n)\) we define two Hessian maps:

\[\begin{aligned} \operatorname{hess}(f) = \left(\frac{\partial ^2 f}{\partial x_i\partial x_j}\right)_{i,j=0,...,x_n} \end{aligned}\]

and

\[\begin{aligned} \operatorname{Hess}(f) = \operatorname{det} \operatorname{hess}(f). \end{aligned}\]

If \(f\in \operatorname{Sym}^d(V)\) then \(\operatorname{Hess}(f)\in \operatorname{Sym}^{(d-2)(n+1)}V\).

Theorem: (Gordan-Noether) On \(\mathbb P^1\), \(\mathbb P^2\), and \(\mathbb P^3\) and a regular function \(f\) on one of these spaces (so a homogeneous polynomial \(f\)), the Hessian vanishes \(\operatorname{Hess}(f) = 0\) if and only if \(V(f)\) is a cone. On \(\mathbb P^n\) for \(n\geq 4\), there exist homogeneous \(f\) such that \(V(f)\) is not a cone and \(\operatorname{Hess}(f) = 0\).

Consequences of Gordan-Noether

One immediate consequence is that the general fiber of the Gauss map is a linear space. The Gauss map is a rational map \(\mathbb P(V)\xrightarrow{\nabla f} \mathbb P(V^\vee)\) where \(f\) is some homogeneous polynomial. It is also a rational map from \(X = V(f)\subset \mathbb P(V) \to X^\vee\subset \mathbb P(V^\vee)\)

Theorem: (Dimca-Papadima, 2001)

\[\begin{aligned} \deg(\nabla f) = (-1)^n\chi\big(\mathbb P(V)\setminus (V(f) \cup H)\big) \end{aligned}\]

where \(H\) is a general hyperplane.

Talk 4: Local heights on hyperelliptic curves for quadratic Chabauty

\(C\) will be a nice curve (smooth, projective, other stuff) over \(\mathbb Q\). The goal is to understand rational points of \(C\), \(C(\mathbb Q)\).

\(g\) is the genus, \(J\) is the Jacobian of \(C\). It turns out that \(J(\mathbb Q)\) is a finitely generated abelian group:

\[\begin{aligned} J(\mathbb Q) = \mathbb Z^r + \text{Torsion}. \end{aligned}\]

We call \(r\) the Mordell-Weil rank.

Earlier results

Falting's theorem (1981): If \(g > 2\), then \(\# C(\mathbb Q) < \infty\).
Chabauty (1941) and Coleman (1985): If \(g\geq 2\) and if \(r < g\), then \(\# C(\mathbb Q) < \infty\) computationally.

Today, we'll assume that \(r = g\).

Heights and quadratic Chabauty

The \(p\)-adic Nekovar height function \(h_z:C(\mathbb Q)\to C(\mathbb Q)\) decomposes as a sum of local heights:

\[\begin{aligned} h_Z(Q) = \sum_{\ell \text{ prime}} h_{Z,\ell}(Q) \end{aligned}\]

where \(h_{Z, \ell}: C(\mathbb Q_\ell) \to \mathbb Q_p\).

Facts:

For \(\ell \neq p\), the local height function \(h_{Z,\ell}\) takes only finitely many values. Remember that \(h_{Z,\ell}\) maps from \(C(\mathbb Q_\ell)\), so we're looking at potentially infinitely many points.
If \(\ell \neq p\) of potential good reduction, then \(h_{Z,\ell} = 0\). This occurs for all but finitely primes.
\(h_{Z,\ell}\) can be defined using intersection pairings on a regular model of \(C\).
The local height function at \(\ell = p\) is well understood: \(h_{Z,p}\) is a locally analytic function.

Quadratic Chabauty

Fix a basepoint \(b \in C(\mathbb Q)\). \(Z \subset C\times C\) is a trace \(0\) correspondence on \(C\) fixed by the Rositi involution. Not clear what this means.

Theorem: (Balakrishnan-Dogra 2018). Assume \(r = g\). Let \(p\) be a prime of good reduction. Then there exist a quadratic function \(\eta_Z:\Lie(J_{\mathbb Q_p}) \to \mathbb Q_p\) for which \(C(\mathbb Q)\) is contained in the locus inside \(C(\mathbb Q_p)\) cut out by the equation \(h_Z(Q) - h_{Z,p}(Q) \in \Omega\) where

\[\begin{aligned} \Omega = \left\{\sum_{\ell\neq p}h_{Z,\ell}(Q_\ell) ~\middle | ~Q_\ell \in C(\mathbb Q_{\ell})\right\}. \end{aligned}\]

It is unknown how to compute this \(\Omega\) set in general, but for specific curves or for classes of curves (i.e. hyperelliptic) it is possible.

Theorem: Let \(C/\mathbb Q\) be a hyperelliptic curve \(y^2 = f(x)\) of some genus \(g\geq 2\). Then there is an explicit, practical combinatorial method for computing \(h_{Z,\ell}\) for all \(\ell \neq p, 2\).

Talk 5: KSBA moduli spaces of stable varieties

Theorem: (Main Theorem.) The KSBA moduli space of stable log Calabi-Yau surfaces, up to a finite cover, is a toric variety.

The reason this is surprising is that there is no underlying toric assumption. Proof will come from mirror symmetry, used to compactify moduli spaces.

Definition: \(\overline{\mathcal M}_{g,n}\) is the moduli space of genus \(g\) "stable curves" with \(n\) marked point \((C, p_1,...,p_n)\)

at worst nodal singularities
stability condition: \(\# \operatorname{Aut}(C) < \infty\iff K_X + \sum p_i\) is ample

Definition: "Stable pairs" \((Y,B)\) where \(Y\) is a projective variety, \(B\) is a \(\mathbb Q\)-divisor.

\((Y,B)\) has semi-log canonical singularities
\(K_Y + B\) is \(\mathbb Q\)-Cartier and ample. \(\mathcal M^{KSBA}\) is the moduli space of stable pairs.

Take \(P\) to be a lattice polytope in \(\mathbb R^n\), this gives us a polarized projective toric variety.

\(Y = \operatorname{Proj}\mathbb C[C(P)(_{\mathbb Z}]\)
\(D = -K_Y\) is the toric boundary divisor
\(L\) is an ample line bundle, \(H^0(Y,L)\cong \bigoplus_{p\in P_{\mathbb Z}}\mathbb Cz^p\).
\(\mathcal M_{(Y,D,L)}\) is the closure in \(\mathcal M^{KSBA}\) of the locus of stable pairs \((Y,D+\epsilon H)\) where \(H\in |L|\) and \(0<\epsilon<<1\).

Theorem: (Alexeev, 2002). For a toric pair \((Y,D)\) with an ample line bundle \(L\) the (normalization of the) moduli space \(\mathcal M_{(Y,D,L)}\) is a toric variiety with associated fan given by the secondary fan of the momentum polytope of \((Y,D,L)\).

Note that a "polarized algebraic variety" is one with a choice of ample line bundle \(L\) in the Neron-Severi group.

What does "stability" mean for a pair \((Y,D,L)\)? (I missed it)

Very hard to take notes during this talk, it was a slide talk.

Talk 6: Hannah Larson, Cohomology of Moduli Spaces of Curves

Joint work with Samir Canning and Sam Payne.

\(\mathcal{M}_g\) is the moduli space of smooth projective genus \(g\) curves. \(\mathcal{M}_{g,n}\) is the moduli space of smooth projective genus \(g\) curves with \(n\) marked points.

Fact: \(\mathcal M_{1,1} \xrightarrow{\sim}{j} \mathbb C\), i.e. elliptic curves over \(\mathbb C\) are classified by their \(j\)-invariant.

Definition: A pointed curve \((C, p_1,...,p_n)\) is stable if \(p_i\in C^{smooth}\), \(C\) has at worst nodal singularities and

every genus \(0\) component of the normalization has \(\geq 3\) special points (nodes and marked points) and
every genus \(1\) component of the normalization has \(\geq 1\) special point.

We write \(\overline{\mathcal{M}}_{g,n}\) to mean the Deligne-Mumford compactification of \(\mathcal{M}_{g,n}\), i.e. add in the stable nodal curves.

These definitions hold over any field, but we're taking \(\mathbb C\) to be our field whenever we draw a picture.

Picture of \(\overline{\mathcal M}_{2,0}\)

Here we see a cartoon of the boundary of \(\overline{\mathcal M}_{2,0}\). One thing to highlight is that, as we "move deeper into the boundary", our curves get simpler until we eventually have two singular genus 0 components glued together at a node, but the combinatorics of the singularities grow more complicated.

Question: What is \(H^*_{\mathbb Q}(\overline{\mathcal{M}}_{g,n})?\) This is a very hard question to answer, it's the same question Dhurv asked in his colloquium talk. A strategy to study it is to produce interesting classes in the cohomology ring.

Tautological classes

The spaces \((T_pC)^\vee\) fit together into a line bundle \(\omega_\pi\), the relative dualizing sheaf. Here \(\pi:\overline{\mathcal M}_{g,1}\xrightarrow{\pi}\overline{\mathcal M}_{g}\). The \(\psi\)-class of this bundle is a class in \(H^2(\overline{\mathcal{M}}_{g,1})\),

\[\begin{aligned} \psi = c_1(\omega_\pi)\in H^2(\overline{\mathcal{M}}_{g,1}) \end{aligned}\]

and pushing this forward produces classes in \(H^{2j}(\overline{\mathcal{M}}_g)\):

\[\begin{aligned} \kappa_j = \pi_*\psi^{j+1}\in H^{2j}(\overline{\mathcal{M}}_g). \end{aligned}\]

These two classes together with a few more bits of machinery produce tautological classes.

Forgetful pullbacks

Take \(f:\overline{\mathcal{M}}_{g,n}\to \overline{\mathcal M}_g\) to be the map which forgets all marked points and \(f_i:\overline{\mathcal{M}}_{g,n} \to \overline{\mathcal M}_{g,1}\) to be the map forgetting all but the \(i\)th marked point. These give us the following classes:

Set \(\tilde{\kappa_j} = f^*\kappa_j\in H^{2j}(\overline{\mathcal{M}}_{g,n})\)
Set \(\psi_i = f^*\psi \in H^{2}(\overline{\mathcal{M}}_{g,n})\).

Monomials

Taking monomials in \(\tilde \kappa_j\) and \(\psi_i\) give us more classes.

Gluing pushforwards

Maps such as \(\overline{\mathcal M}_{1,1} \times \overline{\mathcal M}_{2,3} \to \overline{\mathcal M}_{3,2}\) gives us a pushforward map which, on monomials, looks like \(\psi_i\otimes \kappa_1\psi_3 \mapsto \text{"decorated stratum classes"}\).

Finally, taking \(\mathbb Q\)-linear combinations of these decorated stratum classes gives us the tautological subring \(RH^*(\overline{\mathcal{M}}_{g,n})\subseteq H^*(\overline{\mathcal{M}}_{g,n})\). It lives only in even degree cohomology.

Most classes we can build, Chern classes of bundles, Hodge classes, even the Chern classes of the Verlinde bundle – they're all tautological classes. This begs the question:

Question: For which \((k,g,n)\) do we get \(RH^k(\overline{\mathcal{M}}_{g,n}) = H^k(\overline{\mathcal{M}}_{g,n})\)?

What we know

Equality holds for \(k=0,1,2,3,5\) (Arbarello-Comalba 1990) and \(k = 7,9\) (Bergstroim-Faber-Payne 2022). Sam Payne also showed that \(H^{11}(\overline{\mathcal M}_{1,11}) \neq 0\), and hence equality doesn't hold in general.

Theorem: (Canning-L-Payne). \(H^{11}(\overline{\mathcal{M}}_{g,n}) = 0\) unless \(g=1\) and \(n\geq 11\), in which case it is generated by pullbacks along \(\overline{\mathcal{M}}_{g,n} \to \overline{\mathcal M}_{1,11}\).

Definition: The semi-tautological extension (STE) generated by \(\{\alpha \in H^k(\overline{\mathcal{M}}_{g,n})\}\) is the output of running (1)-(4) and starting with \(\tilde \kappa\), \(\psi\) and \(\alpha\).

Theorem: (Canning-L-Payne). \(H^{13}(\overline{\mathcal{M}}_{g,n})\) lies in the STE generated by \(H^{11}(\overline{\mathcal M}_{1,11})\).

Theorem: (Canning-L-Payne). \(H^*(\overline{\mathcal{M}}_{g,n}) = RH^*(\overline{\mathcal{M}}_{g,n})\), and this implies that \(H^6(\overline{\mathcal{M}}_{g,n})=RH^*(\overline{\mathcal{M}}_{g,n})\) for \(g\geq 10\). For even \(k\leq 14\), \(H_k(\overline{\mathcal{M}}_{g,n}) = RH_k(\overline{\mathcal{M}}_{g,n})\). Finally, \(H_{15}(\overline{\mathcal{M}}_{g,n})\) lies in the STE generated by \(H^{11}(\overline{\mathcal M}_{1,11})\) and \(H^{15}(\overline{\mathcal M}_{1,15})\).

Talk 7: Alicia Lamarche. Wonderful compactifications, toric varieties and derived categories

Motivation: Understand \(D^b(\textsf{Coh} X)\), the derived category, where \(X\) is a smooth variety over \(\mathbb C\). Can treat \(D^b(X)\) as complexes of finitely generated \(R\)-modules.

Two nice things: Semiorthogonal Decompositions (SOD) and Full Exceptional Collections (FEC).

Beilinson showed that \(D&b(\mathbb P^n)\) can be decomposed as \(\mathcal O, \mathcal O(1),...,\mathcal O(n)\).

Theorem: (Kawamata, 2006). Smooth toric varieties have FEC.

There's a moduli space called \(\overline{\mathcal {LM}}_n\) which is a toric variety. It comes from roots systems \(A_n\).

Theorem: \(S_n\) equivariant FEC for \(\overline{\mathcal{LM}}_n\).

\[\begin{aligned} D^b(\overline{LM}_N) = \left\langle D^b_{cusp}(\overline{LM}_N), \{\pi^*_k D^b_{cusp}(\overline{LM_{N/K}})\}_{K\subset N}, \mathcal O\right\rangle. \end{aligned}\]

\(N = \{1,...,n\}\), \(\pi_K:\overline{\mathcal{LM}}_N \to \overline{\mathcal{LM}}_{N\setminus K}\) is the forgetful map. \(D^b_{cusp} = \text{objects } E \text{ such that } R\pi_iE = 0\) where \(1\leq i\leq n\).

Brion noticed some line bundles that generate \(D^b_{cusp}\) come from wonderful compactifications.

Talk 8: David Eisenbud. Infinite Resolutions.

A lot of the effort in the study of infinite resolutions has gone towards constructing them. There are two main cases that people know things about.

The residue field case

Let \(S = k[x_0,...,x_n]\) and look at a homogeneous ideal \(I\subset S\) inside a maximal ideal \(\mathfrak m\). Set \(R = S/I\) and begin to resolve the residue field \(k = R/\frakm\) via \(R\). This is infinite.

Wild conjecture: With \(S\), \(R\) and \(I\) above, \(I_1(F.dd_i) + I_1(F.dd_{i+1})\) becomes constant. This is Macaulay2 notation, \(I_1\) means the \(1\times 1\) minors of the matrix \(F.dd_{i}\).