Boston College Algebraic Geometry Seminar

Abstract: In 1966, Tate proposed the Artin–Tate conjectures, which describe the special values of zeta functions associated to surfaces over finite fields. Building on this, and assuming the Tate conjecture, Milne and Ramachandran formulated and proved analogous conjectures for smooth proper schemes over finite fields. However, the formulation of these conjectures already relied on other unproven conjectures. In this talk, I will present an unconditional formulation and proof of these conjectures. The approach relies on the theory of F-Gauges, a notion introduced by Fontaine–Jannsen and further developed by Bhatt–Lurie and Drinfeld, which has been proposed as a candidate for a theory of p-adic motives. A central role is also played by the notion of stable Bockstein characteristics, which will be introduced in the talk.    

Abstract: In this talk, I apply Koszul duality to the Fukaya A-infinity algebra of a compact Lagrangian $L$ to study questions in mirror symmetry. 

The focus is on a localized mirror constructed intrinsically from the Floer theory of $L$. This construction leads to a natural equivalence between the local Fukaya category generated by $L$ and the category of finite-dimensional representations of the quiver algebra Koszul dual to $CF(L,L)$. Under suitable finiteness conditions, I explain how this local mirror can be globalized to recover the full mirror.

Mumford famously asked for a description of the ample cone of the moduli space of Deligne-Mumford stable curves $\overline{M}_{g,n}$, or, dually, its Kleiman-Mori cone of curves. Conjecturally, the answer is given by the well-known F-conjecture, which states that the cone of curves is spanned by the one-dimensional boundary strata called F-curves.

A breakthrough Bridge Theorem by Gibney, Keel, and Morrison reduces the positive genus case to the case of $g=0$. In genus 0, the F-conjecture for $\overline{M}_{0,n}$ is implied by the strong F-conjecture, which is known to be true for $n \leq 7$ but fails for $n \geq 12$ by Pixton's counterexample.

I will report on joint work with Anton Mellit in which we prove the strong F-conjecture for $n=8$, the F-conjecture for $\overline{M}_{g}$

for $g \leq 44$, and related results. (Conceptually, one needs to solve $2^{2^n}$ linear optimization problems to do this; we prune this

to roughly $1.3\times 10^8$ LP problems, which we solve in a reasonable time using an HPC cluster such as Boston College’s Andromeda 2.)

After surveying the results of the last 20 years on the structure of effective divisors on $\overline{M}_{0,n}$, we show that the pseudo-effective cone of divisors is not polyhedral for $n\geq8$. Using ideas from Teichmüller dynamics and birational geometry, we construct an extremal non-polyhedral ray of the dual cone of moving curves using residue maps of strata of meromorphic differentials.

Logarithmic Gromov-Witten theory, developed by Abramovich, Chen, Gross and Siebert, offers a route to studying the enumerative geometry of log smooth pairs (X,D) using log stable maps. A property satisfied by the theory discovered by Abramovich and Wise is that it has controlled behavior under strata blowups, meaning the enumerative invariants of a log smooth pair (X,D) are also enumerative invariants of a strata blowup (X',D'). I will discuss upcoming work which gives an extension of this result and previous work of mine in the setting of punctured log maps to give a formula for arbitrary enumerative invariants of (X',D') in terms of enumerative invariants of (X,D). As an application, we will show the punctured log Gromov-Witten classes of a split toric bundle over a log smooth base may be expressed in terms of the punctured log Gromov-Witten classes of the base. In particular, the classes of the former theory yield tautological classes in the moduli space of curves if and only if the classes of the latter theory do. Important input to this expression is an alternate description and extension of the double ramification cycle with target involving moduli spaces of punctured log maps.

We define a stability notion for weighted degree 10 hypersurfaces in the weighted projective space P(1,2,3,5), and, more generally, for (10,5) intersections in P(1,2,3,5,5), termed CM stability. We obtain explicit descriptions for these surfaces and prove that the stack of CM semistable (10,5) intersections in P(1,2,3,5,5) admits a proper good moduli space, yielding a compactification of the moduli space of quasi-smooth weighted degree 10 hypersurfaces in P(1,2,3,5).