At the poster session Saturday evening, November 9th. It will start at 6pm in the Loft in LCB.
Rahul Ajit (Utah)
Title: Canonical module and Test module of (Extended) Rees Algebra
Abstract: For an excellent ring of arbitrary charecteristic, we show that, for any ideal, canonical modules of Rees and extended Rees algebras agree in positive degree. Using this, we give decompositions of corresponding multiplier and test modules. As an application, we answer a conjecture of Hara-Watanabe-Yoshida concerning F-Rationality of these in full generality. This is joint work with Hunter Simper. Finally, we use our decomposition to compute test ideal for non-principal ideals in Macaulay2. The Macaulay2 package is additionally joint work with Karl Schwede and Trung Chau.
Tai-Hsuan Chung (UCSD)
Title: Stable Reduction via the Log Canonical Model
Abstract: We formulate a stable reduction conjecture that extends Deligne--Mumford's stable reduction to higher dimensions and show, through a simple proof, that it holds in sufficiently large characteristic, assuming two conjectures. As a consequence, we recover the Hacon--Kovács theorem on the properness of the moduli stack $\overline{\mathscr{M}}_{2,v,k}$ of stable surfaces of volume $v$ defined over $k=\overline{k}$, provided that $\operatorname{char}k>C(v)$, a constant depending only on $v$.
Haohua Deng (Duke)
Title: Hodge-theoretic completion of period maps
Abstract: Half a century ago, Griffiths proposed the question on finding completion of general period maps with significant geometric and Hodge-theoretic meanings. I will show some recent breakthroughs on this project, as well as its applications in moduli theory.
Joshua Enwright (UCLA)
Title: Log Calabi-Yau pairs of complexity zero and arbitrary index
Abstract: Brown-McKernan-Svaldi-Zong proved a conjecture of Shokurov characterizing toric varieties in terms of an invariant of log pairs called the complexity. A special case of their results imply that a log Calabi-Yau pair of index one and complexity zero is a toric log Calabi-Yau pair. We generalize this result to log Calabi-Yau pairs of arbitrary index by showing that a log Calabi-Yau pair of complexity zero is a “toric boundar arrangement.” As an application, we show that any log Calabi-Yau pair of birational complexity zero is crepant to a toric boundary arrangement supported on a generalized Bott tower. This generalizes a result of Mauri-Moraga stating that a log Calabi-Yau pair of index one and birational complexity zero is crepant to the projective space with its toric boundary.
This is joint work with Fernando Figueroa.
Fernando Figueroa Zamora (Northwestern)
Title: Algebraic Tori in the complement of quartic surfaces
Abstract: Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Previously Ducat showed that all coregularity 0 log Calabi-Yau pairs $(\mathbb{P}^3,S)$ are crepant birational to a toric model. A stronger property to ask for is for the complement of $S$ to contain a dense algebraic torus. In this work we start the classification of coregularity zero, slc quartic surface for which their complements contain a dense algebraic torus.
We fully classify the case of reducible surfaces. In the case of irreducible surfaces we are able to classify the cases where the singular locus is non planar.
This is based on joint work with Eduardo Alves da Silva and Joaquín Moraga
Jaime Ignacio Negrete González (Georgia)
Title: Classification of Horikawa surfaces with T-singularities
Abstract: We classify all projective surfaces with only T-singularities, K2=2pg−4, and ample canonical class. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Kollár--Shepherd-Barron--Alexeev (KSBA) moduli space of Horikawa surfaces. We also prove that they are not smoothable when pg≥10, except for the Lee-Park (Fintushel-Stern) examples, which we show to have only one deformation type unless pg=6 (in which case they have two). This demonstrates that the challenging Horikawa problem cannot be addressed through complex T-degenerations, and we propose new questions regarding diffeomorphism types based on our classification. Furthermore, the techniques developed in this paper enable us to classify all KSBA surfaces with only T-singularities and K2=2pg−3, e.g. quintic surfaces and I-surfaces. This is based on a joint work with Vicente Monreal and Giancarlo Urzúa
Brian Nugent (U. Washington)
Title: Moduli of Very Ample Line Bundles
Abstract: We construct moduli spaces for very ample line bundles on a variety X. There is a moduli space of all line bundles on a variety known as the Picard scheme but the locus of very ample line bundles within the Picard scheme is, in general, not even locally closed so one cannot give it the structure of a subscheme. We show how to make a reasonable change to the moduli functor that results in a fine moduli space VA(X) that parametrizes “nice” families of very ample line bundles. This has connections to Brill-Noether theory for higher dimensional varieties.
Sridhar Venkatesh (Michigan)
Title: The intersection cohomology Hodge module on toric varieties
Abstract: The intersection cohomology complex IC_X on a toric variety X has been well studied starting with the works of Stanley and Fieseler, and more recently, the works of de Cataldo-Migliorini-Mustata and Saito. However, it has a richer structure as a Hodge module (denoted IC^H_X) in the sense of Saito’s theory, and so we have the graded de Rham complexes gr_k(DR(IC^H_X)), which are complexes of coherent sheaves carrying significant information about X. We will describe the generating function of the cohomology sheaves of gr_k(DR(IC^H_X)) and give a precise formula relating it with the stalks of the perverse sheaf IC_X (in particular, this implies that the generating function depends only on the combinatorial data of the toric variety). This is joint work with Hyunsuk Kim.
José Ignacio Yáñez (UCLA)
Title: Polarized endomorphisms of log Calabi-Yau pairs
Abstract: An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. We consider the case of log Calabi-Yau pairs (X,B), and we prove that if (X,B) admits a polarized endomorphism that preserves the boundary structure, then (X,B) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.
Yilong Zhang (Purdue)
Title: An Elliptic-Elliptic Surface with Maximal Picard Number
Abstract: For a smooth algebraic surface over the complex numbers, the Picard number, denoted by $\rho$, is bounded above by the Hodge number $h^{1,1}$. A surface is said to have maximal Picard number when $\rho = h^{1,1}$. Examples of such surfaces are rare, especially when the Kodaira dimension is a least zero. Notable instances include K3 surfaces with Picard number 20, classified by Shioda and Inose via transcendental lattices. For surfaces with Kodaira dimension one, Shioda’s modular surfaces achieve maximal Picard number but have only torsion sections.
In this poster, we present an example of an elliptic surface with Kodaira dimension one, maximal Picard number, and a section of infinite order. This surface has $p_g = q = 1$, and its second cohomology admits a K3-type Hodge structure. As the base is an elliptic curve, this type of surface is referred to as an elliptic-elliptic surface. We will explore its deformation within an explicit Noether-Lefschetz locus, along with the associated period map. Furthermore, we compare its degeneration at boundary points to the Kulikov models of K3 surfaces. This work is a collaboration with Francois Greer.