Abstracts

Aaron Bertram (Utah)

Title: Stability and Regularity

Abstract: In this talk, I want to relate the graded syzygies of a coherent sheaf on projective space with a family of "Euler" stability conditions. I am promoting a point of view: graded syzygies (especially the minimal free resolution) should be organized into filtrations, which will often be Harder-Narasimhan filtrations for appropriate stability conditions. The picture that emerges is a series of moduli spaces, each constructed by Geometric Invariant Theory, that culminates in the moduli space of Gieseker-stable sheaves.

• Matthew Stevenson

  Essential Skeleta of Pairs

  Abstract: To a variety over a non-Archimedean field, one can associate its Berkovich analytification, which is a space of semivaluations on the function field of the variety. When working over a discretely-valued field of residue characteristic zero, Mustaţă—Nicaise introduced a canonical subspace of the Berkovich analytification called the essential skeleton; by work of Nicaise—Xu, the essential skeleton is identified with the dual complex of a minimal dlt-model of the variety, when it exists. We will describe how one constructs the essential skeleton of a pair (or, more generally, of a log regular log scheme) and how this compares with canonical skeleta that exist in the toric and trivially-valued settings. This is joint work with Enrica Mazzon.

• Emanuel Reinecke

  Autoequivalences of twisted K3 surfaces

  Abstract: Autoequivalences of a variety X are natural generalizations of the automorphisms of X. When X is a complex, projective K3 surface, Bridgeland conjectured an alluring description of the group of autoequivalences of X. One part of this conjecture has been established in work of Huybrechts, Macri, and Stellari. In this talk, we explain a new proof of their result. We first review the necessary background on K3 surfaces and derived categories and give a precise definition of the notion of autoequivalences. Then, we construct a natural action of the autoequivalences on the cohomology of X, which is essential in formulating Bridgeland's conjecture. Finally, we demonstrate how passing to the larger class of twisted K3 surfaces helps to prove part of the conjecture.

• Stefano Filipazzi

  Some facts about generalized pairs

  Abstract: Recently, Birkar and Zhang introduced the concept of generalized polarized pair. This kind of pairs, which arise naturally from the canonical bundle formula, play a central role in recent developlments, such as the study of the Iitaka fibration and the proof of BAB conjecture. A natural question is whether these generalized pairs satisfy the nice properties of usual log pairs. In this talk, we will discuss invariance of plurigenera and volumes in this new setup. Then, moving to the case of surfaces, we will address the boundedness of these objects.

• Akash Sengupta

  Manin's conjecture and the Fujita invariant of finite covers

  Abstract: Let $X$ be a Fano variety defined over a number field. Manin's conjecture predicts an asymptotic formula for the number of rational points on $X$ with bounded height. According to the conjecture, the growth of rational points is governed by the Fujita invariant (or the $a$-constant). In this talk we will use birational geometric methods to prove statements related to Manin's conjecture. In particular, we will study the behavior of the Fujita invariant under pull-back to generically finite covers and prove a statement about geometric consistency of Manin's conjecture.

• David Wen

  Towards Minimal Models of Elliptic Fourfolds

  Abstract: Fiber spaces play an important role in the minimal model program as the results can be categorized into Mori fiber space, Iitaka fibrations over canonical models and varieties of general type. A natural problem to consider would be, if we started with an algebraic fiber space, how might it behave with respect to the minimal model program. For case of elliptic threefolds, it was shown by Grassi, that minimal models of elliptic threefolds related to log minimal models of the base surface and showed that minimal models, in a sense, has to respect the fiber structure for elliptic threefolds. I will present ideas towards a generalization to the case of elliptic fourfolds and higher dimensions.

• Michael Perlman

  Regularity of Pfaffian thickenings

  Abstract: Let S be the ring of polynomial functions on the space of n x n complex skew-symmetric matrices. This ring has a natural action of the group GL(n), and the invariant ideals correspond to scheme-theoretic thickenings of the Pfaffian varieties. For each invariant ideal I in S, we compute the modules Ext^i(S/I,S). When the thickening is defined by a power of an ideal of Pfaffians, we use these Ext computations to determine the Castelnuovo-Mumford regularity. This allows us to characterize when powers of ideals of Pfaffians have linear minimal free resolution.

• Yordanka Kovacheva

  Intersection Pairing for Cycles and Determinant Line Bundle

  Abstract: I construct an intersection pairing of cycles modulo relations and the corresponding determinant line bundle. More specifically, I consider the map ${CH^p(X)\times CH^q(X)\to \text{Pic}(S)}$ of Chow groups of a variety $X$ over a base $S$. Here $p+q=d+1$, where $d$ is the relative dimension of the morphism $X\to S$. I treat the Chow groups $CH^p(X)$ as categories with objects cycles of codimension $p$ and morphisms arising from the $Z^p(X,1)$ term in Bloch's complex modulo the image of Tame symbols of $K2$-chains. This pairing coincides with the Knudsen-Mumford determinant line bundle using the structure sheaves of the cycles on $X$. For codimension $q$ cycles that are algebraically trivial on the generic fiber $X_{\eta}$, I show that the image in $\text{Pic}(S)$ does not depend on the rational equivalence of the codimension $p$ cycles. Nevertheless, when working with numerically trivial divisors and zero cycles, the image does depend on the rational equivalence of the zero cycles. As a part of the proof, I construct an explicit isomorphism ${H^1_{et}(X, \mathbb{Z}/n)\to Hom(H_1(Sus_{\bullet}(X)/n), \mathbb{Z}/n)}$, which in the case of smooth curves is the Weil pairing. Equivalently, this is an isomorphism $H_1(Sus_{\bullet}(X)/n)\to \pi_1^{et,ab}(X)/n$. Based on the constructed pairing, I construct a line bundle on ${CH^p_{alg}(X)\times CH^q_{alg}(X)}$, which I want to prove is canonically isomorphic to the pull-back via the Abel-Jacobi map of the Poincare line bundle on the Intermediate Jacobians $J^p(X)\times J^q(X)$. I also hope to extend the pairing to algebraic singular (Suslin) homology and in motivic setting.

• Hongshan Li

  A Vanishing Theorem for Higgs Sheaf

  Abstract: Let $X$ be a smooth projective variety over $\complex$ with an embedding $O_X(1)$, and $D \subset X$ be a simple normal crossing divisor. Suppose $(E, \theta)$ is a Higgs sheaf with trivial \emph{parabolic structure} on $D$ such that 1) $E$ is semistable; 2) $c_i(E) = 0$ for all $i$; 3) $\theta$ is nilpotent. Such a Higgs sheaf arises naturally when taking associate graded sheaf of a polarized Hodge structure with unipotent monodromy at $D$. A result of D.Arapura states that \[ \mathbb{H}^i(X, \text{Gr}_F\text{DR}(V, \nabla)\otimes L) = 0 \] for $i > \dim X$. In this joint work with Donu Arapura and Feng Hao, We will generalize D.Arapura's result via a cyclic cover so that the Higgs sheaf in question could have non-trivial parabolic structure on $D$. Moreover, by using the moduli space of parabolic Higgs sheaf and upper-semicontinuity of cohomology, we can remove the assumption that $\theta$ is nilpotent.

• Noah Winslow

  Title: Obstructions to Low-dimensional Realizations: Most Octahedra are 4-Dimensional

  Abstract: Embed a graph G generically into R^n as a bar framework (edges are rigid straight bars which are free to rotate around vertices). Fixing the edge lengths given by the embedding, what is the smallest integer d such that G can embed into R^d with the same edge lengths? For example, no n-simplex can be generically embedded into R^{n-1}. One can show that non-realizability is a minor closed property so it is natural to ask: Are simplicies the minor minimal objects with regards to realizability? No, we can find generic 4-dim embeddings of the octahedron which do not admit 3-dim realizations. We will demonstrate a proof of this fact and examine several different characterizations of this correspondence between graphs and these embedding obstructions in the search for additional forbidden minors in dimensions larger than 4. This is a work in progress.

• Franco Rota

  Title: Bridgeland stability on orbifolds

  Abstract: Abstract: Most of the work on Bridgeland stability conditions focuses on the derived category of a smooth projective variety, because of technical difficulties in approaching singular varieties. To bypass singularities, we look at stability conditions on the derived category of smooth stacks instead. We construct stability conditions on orbifolds, and focus on the example of a root stack over a curve. In this case, there is an interesting correspondence between wall-crossing on the stability manifold and moduli spaces of parabolic bundles.

• Mao Li

  Title: Construction of the Poincare sheaf on the stack of Higgs bundles

  Abstract: An important part of the Langlands program is to construct the Hecke eignsheaf for irreducible local systems. Conjecturally, the classical limit of the Hecke eignsheaves should correspond to the Poincare sheaf on the stack of Higgs bundles. The Poincare sheaf for the compactified Jacobian of reduced planar curves have been constructed in the pioneering work of Dima Arinkin. In the present work I constructed the Poincare sheaf on the stack of rank two Higgs bundles for any smooth projective curve over the entire Hitchin base, and it turns out to be a maximal Cohen-Macaulay sheaf. This includes the case of nonreduced spectral curves, and thus provides the first example of the existence of the Poincare sheaf for nonreduced planar curves. The method I use is a variant of Drinfeld's construction of automorphic sheaves for $GL(2)$ in the Higgs setting.

• Siddharth Mathur

  Title: Azumaya Algebras and the Resolution Property

  Abstract: Azumaya algebras, are (etale) twisted forms of matrix rings. These objects are of great utility because they give rise to Brauer classes. Fifty years ago, Grothendieck asked whether every cohomological Brauer class has a corresponding Azumaya algebra. This question is still open even for smooth separated threefolds over the complex numbers! One says a scheme (or Algebraic stack) X satisfies the resolution property if every coherent sheaf is the quotient of a vector bundle. The work of Totaro and Gross explains that this property holds iff X admits a very special quotient stack presentation. However, whether or not separated Algebraic stacks have this property remains a difficult question. The goal of our talk will be to explain (1) Why these two questions are deeply intertwined, (2) New results regarding the existence of Azumaya Algebras and (3) How we can use results as in (2) to show large classes of algebraic stacks satisfy the resolution property.

• Daniel Levine

  Title: Weak Brill-Noether for Del Pezzo surfaces

  Abstract: Let $(X,H)$ be a polarized birationally ruled surface with fiber class $F$ such that $H.(K_X + F) < 0$. Walter showed that the moduli space of semistable sheaves of rank greater than $1$ is irreducible for any Chern character. In this setting, it makes sense to talk about the cohomology of a general semistable sheaf of some fixed Chern character. On Hirzebruch surfaces, a complete answer is known due to Coskun and Huizenga. They also compute these groups in some cases for Chern characters with Euler characteristic $0$ on blowups of Hirzebruch surfaces. We extend some of their results for Del Pezzo surfaces. This is work in progress.

• Takumi Murayama

  Title: Fujita-type conjectures and Seshadri constants

  Abstract: If L is an ample divisor on a projective variety X, then K+mL is basepoint-free for large enough m, where K is the canonical divisor. T. Fujita in 1988 conjectured that if X is n-dimensional, then m = n+1 should suffice for basepoint-freeness. To study this conjecture, Demailly introduced Seshadri constants as a way to measure the local positivity of L. While examples of Miranda show that Seshadri constants cannot answer Fujita's conjecture, Seshadri constants can still prove a version of Fujita's conjecture for basepoint-freness at general points. We will present joint work with Yajnaseni Dutta, which exploits a similar strategy involving Seshadri constants to give positive evidence toward Popa and Schnell's relative version of Fujita's conjecture.

• Arda Huseyin Demirhan

  Title: Counting Rational Points of Bounded Height on the Blow-up of the Projective Space

  Abstract: We study the behaviour of the counting function on the blow-up of $\mathbb P^n$ along $\mathbb P^m$. Particularly, we compute the number of rational points of bounded height in the exceptional divisor $E$ and its complement $U$. As a result, we’ll observe that $E$ is accumulating for some ample sheaves.

• Shizhuo Zhang

  Title: Exceptional collections with applications to rationality problems of surfaces and quiver moduli.

  Abstract: Let $X$ be a smooth projective surface with exceptional collection of line bundles of maximal length. I will relate the rationality of surface to the properties of the exceptional collection. I will prove the Orlov's folklore conjecture for small picard rank surface. I will also show that seems different notions on exceptional collection of line bundles in various area mathematics are actually the same, as a result, we prove several conjectures on projective surfaces. Then, we show that a large class of rational surfaces can be realized as quiver moduli coming from strong exceptional collection of line bundles. The work on quiver moduli is a joint work with Xuqiang Qin.