Patricio Gallardo: Revisiting the moduli of Surfaces of General Type via Abelian Covers
Understanding the classification and moduli of complex surfaces of general type has been one of the classical problems in algebraic geometry. In this talk, we describe families of surfaces' invariants and degenerations constructed via abelian covers, as introduced by R. Pardini. This joint work with J. Mukherjee builds upon previous research with L. Schaffler, G. Pearlstein, and Z. Zhang.
Elham Izadi: Hyperkahler manifolds and Lagrangian fibrations
Hyperkahler manifolds are one of the main classes of manifolds appearing in Berger’s classification of holonomy groups of Riemannian manifolds. It is known that for any non-constant map f from a hyperkahler manifold of dimension 2n, the generic fibers of f are either finite or abelian varieties of dimension n. The latter are Lagrangian fibrations. I will discuss some open problems and some results concerning Lagrangian fibrations on hyperkahler manifolds.a
Jonathan Montano: Rational powers, invariant ideals, and the summation formula
Defined by Lejuene-Jalabert-Teissier and Huneke-Swanson, the rational powers of an ideal is a Q-filtration that contains the radical and the integral closure of the powers of the ideal. In this work we describe the rational powers and the Rees valuations of several classes of invariant ideals in terms of polyhedra. This allows us to show a summation formula for rational powers similar to Mustata's formula for multiplier ideals. Moreover, for arbitrary ideals over the complex numbers, we prove a weaker version of this formula that holds for sufficiently large rational numbers. This is joint work with Sankhaneel Bisui, Sudipta Das, and Huy Tai Ha.
Karl Schwede: Singularities in Mixed Characteristic via Alterations
Multiplier ideals and test ideals are ways to measure singularities in characteristic zero and p > 0 respectively. In characteristic zero, multiplier ideals are computed by a sufficiently large blowup by comparing the canonical module of the base and the resolution. In characteristic p > 0, test ideals were originally defined via Frobenius, but under moderate hypotheses, can be computed via a sufficiently large alteration again via canonical modules. In mixed characteristic (for example over the p-adic integers) we show that various mixed characteristic analogs of multiplier/test ideals can be computed via a single sufficiently large alteration, at least when one builds in a small perturbation term. This perturbation is particularly natural from the perspective of tight closure, almost mathematics or the theory of pairs from birational algebraic geometry. Besides unifying the three pictures, this has various applications. This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Kevin Tucker, Joe Waldron and Jakub Witaszek.
Mark Shoemaker: Counting curves in determinantal varieties and a connection to quiver mutations
Suppose X is a smooth projective variety, E and F are vector bundles on X, and M: E —> F is a map of vector bundles. More concretely, M defines a family of matrices {M_x}, parametrized by the points x of the variety X. For a positive integer k, we define the kth determinantal variety of M to be the locus of points x in X for which the linear map M_x has rank at most k. Such varieties give some of the simplest examples of subvarieties of X which are not complete intersections. Determinantal varieties are almost always singular, however there are two natural desingularizations defined using basic concepts from linear algebra. In this talk I will describe a beautiful correspondence, conjectured by physicists, between the number of curves in each of the two resolutions. I will give a sketch of the proof and connect this correspondence to mutations of quiver varieties. This is based on joint work with Nathan Priddis and Yaoxiong Wen.
Javier Gonzales Anaya: Blow-ups of weighted projective planes at a point: Exploring the parameter space of triangles and the MDS property
Castravet and Tevelev's seminal work on the MDS property for $\bar{M}_{0,n}$ sparked a large effort to understand the Cox ring and MDS property for blow-ups of toric varieties at one general point. This problem is already deep and nuanced for projective toric surfaces defined by rational plane triangles (Picard number 1), which is the focus of our talk. We will consider a parameter space of all such triangles and show how it sheds light on the study of the MDS property for these varieties. This parameter space helps explain most known results in the area, and has also led to surprising new results, including examples of such surfaces with a semi-open Kleiman-Mori cone of curves. This is joint work with José Luis González and Kalle Karu.
Yunfan He: Holomorphic anomaly equation for categorical enumerative invariants of elliptic curves
Categorical enumerative invariants(CEI) is a certain type of invariants associated to a cyclic A-infinity algebra and a splitting of its noncommutative Hodge filtration. The conjecture is that when taking an A-infinity algebra Morita equivalent to a Fukaya type category, the CEI will coincide with Gromov-Witten invariants. We will show that, similar to Gromov-Witten invariants, CEI also satisfy the so called holomorphic anomaly equation, in the elliptic curve case. This is based on the joint work with Andrei Caldararu.
Devlin Mallory: Finite F-representation type for homogeneous coordinate rings
Finite F-representation type is an important notion in characteristic-p commutative algebra, and is closely connected to the behavior of differential operators. Despite this, explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings (essentially, those of Calabi–Yau or general-type varieties) will fail to have finite F-representation type, via an analysis of their rings of differential operators. This illustrates a connection between the commutative-algebraic property of FFRT, and the algebro-geometric properties of positivity/negativity of tangent sheaves of varieties. This also provides instructive examples of the structure of the ring of differential operators for non-F-pure rings, which to this point have largely been unexplored. We will also discuss the case of Fano varieties: Recent work has provided non-toric smooth Fano varieties that do have FFRT (Grassmannians Gr(2,n) and the quintic del Pezzo surface). However, it seems unlikely that this will be true for all Fano varieties; we will present conjectural evidence that “in general” smooth Fano varieties will often fail to have FFRT.
Jayan Mukherjee: Projective smoothing of varieties with simple normal crossings
In this article, we discuss an approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and their smoothings via deformation theory of morphisms. As a consequence, we prove results on the existence and smoothing of snc subvarieties $V \subset \mathbb{P}^N$, with two irreducible components, each of which are Fano varieties of dimension $n>2$, embedded inside $\mathbb{P}^{N}$ for effective values of $N$, by the complete linear series of a line bundle $H$. The general fibers of the resulting one parameter families are either smooth Fano, Calabi-Yau or varieties of general type, depending on the positivity of the canonical divisor of their intersections. An interesting consequence of projective smoothing is that it gives a smoothing of the semi-log-canonical (slc) pair $(V, \Delta)$, where $\Delta = cH$, $c < 1$, is a rational multiple of a general hyperplane section of $H$. For threefolds, we are able to give explicit descriptions of the smoothable snc subvarieties due to the classification results of Iskovskikh-Mori-Mukai. In particular, we show the existence of unions V = $Y_1 \bigcup_D Y_2 \subset \mathbb{P}^N$, where $Y_i$'s are smooth anticanonically (resp. bi-anticanonically) embedded Fano threefolds, intersecting along $D$, where $D$ is either a del-Pezzo surface or a $K3$ surface (resp. a smooth surface with ample canonical bundle) and their smoothing in $\mathbb{P}^N$ to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of $N$ between $10$ and $163$. This is joint work with P. Bangere and F.J. Gallego.
Sarah Poiani: The Duality of Pair Operations
Expanding on the work of Kemp, Ratliff and Shah, for any closure cl defined on a class of modules modules over an Noetherian ring, we develop the theory of cl-prereductions of submodules. For any interior i on a class of $R$-modules, we also develop the theory of i-postexpansions. Using the duality of Epstein, R.G. and Vassilev, we show that if i is the interior dual to cl, then these notions are in fact dual to each other. We further the thematic notion of duality and seek to understand how it arises in the context of properties pair operations (a generalization of closure and interior operations) can be endowed with. In particular, we will see that the notions of order reversing and involutive are self dual, and that independence is dual to spanning. Because involutive and idempotent are strong requirements, we also extend our analysis to the weaker notions of pre- and post- involutive and pre- and post-idempotent ($L\subseteq p^2(L,M)$, $p^2(L,M)\subseteq L$, $p(L,M)\subseteq p^2(L,M)$, and $p^2(L,M)\subseteq p(L,M)$, respectively).