Schedule
Schedule
The talks and professional development will be in Nutrien 140, and the poster sessions will be in the atrium of the Smith Natural Resources Building. See the page Afternoon Information for information on where individual in-person groups meet in the afternoons.
We plan to stream the morning talks live on Zoom. If you want to make sure to be send the link shortly before the first talk begins, please go ahead and fill out the following form: https://forms.gle/cgMXbnAj6wxcnFGGA
Title: K-stability and K-moduli spaces
Abstract: We will introduce the notion of K-stability and outline the key consequences of this stability that yield a well-behaved moduli space of Fano varieties and log Fano pairs. We will also introduce wall-crossing in K-moduli to survey several examples of K-moduli spaces, along with current research directions in the area.
Title: Decomposition theorem of algebraic maps and applications
Abstract: The decomposition theorem of Beilinson, Bernstein, Deligne, and Gabber is a power tool in understanding topology of proper maps. It can be viewed as a relative Hodge theory, extending Deligne’s result on for smooth and proper maps. Remarkably, de Cataldo and Migliorini showed that the decomposition theorem is indeed a consequence of the more classical Hodge theory. In this talk I will review these ideas, and discuss some applications of the decomposition theorem in the study of moduli spaces in classical algebraic geometry, including moduli of curves and sheaves.
Title: Cluster geometry
Abstract: In this talk, we will introduce the concept of cluster type varieties. This concept generalizes toric varieties and spectra of cluster algebras. We will explain how these varieties arise naturally when studying the birational geometry of Calabi-Yau pairs. We will discuss some results regarding cluster type surfaces, cluster type threefolds, and the constructibility of the cluster type property.
Title: A Gentle Introduction to Characteristic p Geometry
Abstract: My talk will be an introduction to characteristic p geometry, intended to spark interest in those who are not yet familiar with or passionate about the subject. We will discuss the historical origins of the study of varieties over fields of positive characteristic and explore some of the many interactions this area has with other domains of mathematics. We will encounter pathological and unexpected behaviors that simply cannot occur in the world of complex geometry and use these to motivate some of the key tools and ideas of the field. Finally, we will touch on a few contemporary research directions and open questions.
Title: Singularities in positive and mixed characteristic
Abstract: In my talk, I will give a brief overview of the theory of singularities in positive and mixed characteristic, in the context of birational geometry and commutative algebra. I will talk about Frobenius splittings and splinters, and how they can be studied using tools such as local cohomology.
Title: Tangent and normal bundles of curves in projective space
Abstract: Various geometric properties of projective varieties X \subset P^r, such as their deformation theory, are encoded in natural vector bundles, such as the restricted tangent bundle T_{P^r}|_X and the normal bundle N_{X/P^r}. In this talk we will explain the significance of these vector bundles, and survey recent results concerning their structure when X is a general curve.
Title: Rationality of algebraic varieties
Abstract: The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that admit a 1-to-1 parametrization by projective space on a dense open subset; hence, rational varieties are the easiest varieties to understand. Historically, rationality problems have been of great importance in algebraic geometry; for example, Severi was interested in finding rational parametrizations for moduli spaces of curves, and the classical Lüroth problem was concerned with determining the rationality of certain varieties. Classically, the rationality problem was first studied over the field of complex numbers. Over fields that are not algebraically closed, the arithmetic of the field adds additional subtleties to the rationality problem. This talk will survey some results on rationality of algebraic varieties, both over ℂ and over other fields.
Title: Arithmetic structures on cohomology of algebraic varieties
Abstract: Originally motivated by the problem of counting points on varieties over finite fields, Grothendieck introduced the theory of étale cohomology of algebraic varieties. Even for a variety over complex numbers this theory endows its usual topological cohomology with an interesting additional structure of the action of the Galois group of its field of definition. I will describe several applications of this structure to problems of classifying algebraic varieties and maps between them, as well as some enhancements of this structure coming from p-adic Hodge theory.
Title: Intersection theory on M_g
Abstract: The intersection theory of M_g has been an active area of research since Mumford's 1983 paper where he introduced the tautological ring. Roughly speaking, tautological classes are classes in the Chow ring that we know must be there by the very definition of the moduli problem. In this talk, I'll give an overview of some key results about the Chow ring of M_g, including what is known about the structure of the tautological ring, when tautological classes generate the entire Chow ring, and recent investigations of non-tautological classes.
Title: Counting solutions to systems of polynomial equations and (matroid) Hodge theory
Abstract: In this talk I will explain how invariants of graphs (and matroid) can be obtained by counting the number of solutions to systems of polynomial equations, and how (matroid) Hodge theory gives inequalities between these graph (and matroid) invariants.
Title: Stability conditions and moduli spaces on noncommutative varieties.
Abstract: In his celebrated work, Mumford introduced the notion of slope stability for vector bundles on curves, leading to the construction of projective varieties parametrizing semistable bundles of fixed rank and degree. Since then, these results have been generalized to higher dimensional varieties and coherent sheaves with outstanding applications in algebraic geometry and differential geometry.
A further breakthrough is due to Bridgeland, who considered the more abstract context of triangulated categories and defined the concept of stability for objects therein. In fact, this notion of stability recovers slope stability when considering the bounded derived category of a smooth projective curve.
The goal of this talk is to give a gentle introduction to the recent progresses in the theory of stability conditions and moduli spaces with particular focus on the applications in hyperkahler geometry.
Title: Compactifying moduli spaces of K3 surfaces
Abstract: Due to Torelli theorems, moduli spaces of K3 surfaces are orthogonal Shimura varieties. In the 60’s-80’s, compactifications of such varieties were constructed by Baily-Borel, Ash-Mumford-Rapaport-Tai, and Looijenga. But are any of these "toroidal" or "semitoroidal" compactifications distinguished, in the sense that they parameterize some notion of stable K3 surfaces, like the Deligne-Mumford compactification for curves? Work on the Minimal Model Program from the 80’s-00’s by Kollar-Shepherd-Barron and Alexeev proved that an ample divisor on a Calabi-Yau variety defines a notion of stability, leading to compact moduli spaces. I will discuss the interaction between the above Hodge-theoretic and MMP approaches to compactification.
Title: Conformal blocks in algebraic geometry
Abstract: Conformal blocks are vector spaces that naturally arise in the context of conformal field theory and depend on two inputs: a representation theoretical one (e.g. a category of representations) and a geometric one (e.g. a pointed curve). In the special case when the representation theoretical input comes from a Lie algebra, conformal blocks are related to the moduli space of principal bundles on a curve. On the other hand, by varying the geometric datum, conformal blocks naturally define sheaves on moduli spaces of pointed curves. Although recent progress has been made in the understanding the properties of these sheaves (and their connections to moduli of bundles), there are still many open questions that remain open and that I will discuss in this talk.
Title: Derived methods and trace theories via the deformational Hodge conjecture.
Abstract: Given a smooth projective morphism $ f \colon X \to S:= \Spec F [\![t]\!] $ ($ F $ a finitely-generated field over $ \mathbb{Q} $) and a class $ \xi $ in $ K_0\left(X \times_{S}\{ 0 \} \right)$, when does there exist a compatible family $ (\xi_n)_{n \geq 0} $ in $ K_0\left(X \times_S \Spec F[t]/t^n\right) $ extending $ \xi $? A necessary condition is that its image $ \mrm{ch}(\xi) $ under the Chern character admits a compatible family of extensions--which we refer to as an `deformational' extension--satisfying a Hodge-theoretic condition. In this talk, we will show that this condition is sufficient, following ideas of Bloch--Esnault--Kerz and Morrow.
Our discussion will take a detour through derived algebraic geometry and trace methods: A result of Goodwillie implies that the existence of an deformational extension in $ K_0 $ is equivalent to the existence of an deformational extension in derived de Rham cohomology. Then, a crucial calculation identifies a derived projective system with its non-derived counterpart.
Title: What is a virtual class?
Abstract: Come to the talk and find out! But, for those of you who like spoilers, here's the most basic example: a scheme (or stack) X inherits a virtual cycle whenever it is realized as a subscheme of Y cut out by a section of a vector bundle E. In this case, the virtual cycle is just the localized euler class of E (as in Fulton's book, section 14.1). In particular, if the section defining X is regular, the associated virtual class is the usual fundamental class of X. For this reason virtual classes are a natural substitute for fundamental classes, and in fact the substitute can be better behaved. For example, virtual classes are of pure dimension even when X has components of many dimensions. Many moduli stacks carry a natural virtual class (arising from incarnations as derived stacks), making virtual classes an essential ingredient in Gromov-Witten and other enumerative theories.
Aitor Lopez (ETH): Noether-Lefschetz cycles on A_g
Abelian varieties with elliptic sub factors give rise to cycles on the moduli space of abelian varieties. We describe its intersection theoretic properties: They are usually non-tautological, generate cycle valued modular forms and have an interesting intersection with the locus of Jacobians. This leads to predictions for Gromov-Witten invariants of a family of elliptic curves.
Amy Q. Li (UT Austin): Vanishing H^1 of the Hurwitz space of degree-3 covers
We compute the first cohomology group of the Hurwitz space of degree 3 covers using an inductive method which relies on the stratification of the boundary of the Hurwitz space. In order to compute the base cases, we use monodromy representations of covers and the monodromy of the target map from the Hurwitz space to $\Mbar_{0,4}$, the moduli space of 4-pointed, genus-0 curves.
Anne Fayolle (Utah): Centers of perfectoid purity
We introduce a mixed characteristic analog of log canonical centers in characteristic 0 and centers of F-purity in positive characteristic, which we call centers of perfectoid purity. We show that their existence detects (the failure of) normality of the ring. We also show the existence of a special center of perfectoid purity that detects the perfectoid purity of R, analogously to the splitting prime of Aberbach and Enescu, and investigate its behavior under étale morphisms.
Charlie Wu (Toronto): Compact components of character varieties
Let $G$ be a real reductive group, and $\Sigma_{g,n}$ an orentiable topological surface of genus $g$ with $n$ punctures. We classify compact components of (relative) character varieties of $G$-representations of $\pi_1(\Sigma_{g,n})$ in terms of Hodge-theoretic data. When $g > 0$, we show that compact components exist only when $G$ is compact. When $g = 0$, we find and study many compact components.
Daebeom Choi (Penn): Extremal effective curves and non-semiample line bundles on the moduli space of pointed curves
We develop a new method for establishing the extremality of curves in the moduli space of curves and determine the extremality of many boundary $1$-strata. As a consequence, by extending Keel’s argument using a general criterion for non-semiampleness, we demonstrate that a substantial portion of the nef cone of $\M{g}{n}$ is not semiample. In particular, we construct an explicit example of a non-contractible extremal ray on $\M{3}{n}$. Our method relies on two main ingredients: (1) the construction of a new and effective collection of nef divisors on $\M{g}{n}$, and (2) the identification of a tractable inductive structure on the Picard group, arising from Knudsen’s construction.
Daniel Mallory (Northwestern): On the K-stability of blow-ups of projective bundles
Originally introduced by differential geometers as a criterion for the existence of Kähler-Einstein metrics on Fano manifolds, K-stability has recently been of great interest to algebraic geometers, as it gives a good moduli theory for Fano varieties. We investigate the K-stability of certain blow-ups of projective bundles over a Fano base. Specifically, for a Fano variety $V$ of dimension $n$, we consider the blow-up $Y$ of the projective compactification of a line bundle $L such that $r L ~ -K_V$ along the image of a divsor $B ~ k L$ along a positive section of the projective bundle. We show that if $V, B$, and $Y$ are smooth, when $k = 2$, the K-stability of $Y$ is equivalent to the K-stability of the pair $(V,aB)$ for some explicit constant $a$ depending on $n$ and $r$, and for $k$ not equal to $2$, $Y$ is K-unstable.
Erin Dawson (Colorado State): Tropical Tevelev degrees
Tropical Hurwitz spaces parameterize genus g, degree d covers of a tropical rational curve with fixed branch profiles. Since tropical curves are metric graphs, this gives us a combinatorial way to study Hurwitz spaces. Tevelev degrees are the degrees of a natural finite map from the Hurwitz space to a product Mgnbar{g,n} cross Mgnbar{0,n}. In 2021, Cela, Pandharipande and Schmitt presented this interpretation of Tevelev degrees in terms of moduli spaces of Hurwitz covers. We define the tropical Tevelev degrees, \Tev_g^\trop in analogy to the algebraic case. We develop an explicit combinatorial construction that computes \Tev_g^\trop = 2^g$.
Feiyang Lin (UC Berkeley): Resolving the singularities of splitting loci
We construct modular resolutions of singularities for splitting loci, and use them to show that tame splitting loci have rational singularities. As a corollary of our results and Hurwitz-Brill-Noether theory, we prove that if C is a general k-gonal curve, the components of W^r_d(C) have rational singularities. We also recover the classical Gieseker-Petri theorem. Along the way, we prove a cohomology vanishing statement for certain tautological vector bundles on Quot schemes on P1, which may be of independent interest.
Francesca Rizzo (IMJ-PRG): On the fixed locus of the antisymplectic involution of an EPW cube
EPW cubes are polarized hyper-Kähler varieties of K3[3]-type that carry an anti-symplectic involution. We study the geometry of the fixed locus $\sW_A$ of this involution and prove that it is a \emph{rigid} atomic Lagrangian submanifold. Our proof is based on a detailed description of certain singular degenerations of EPW cubes and the degeneration methods of Flappan--Macrì--O'Grady--Saccà.
Hao Zhang (Glasgow): Gopakumar--Vafa invariants associated to $cA_n$ singularities
This paper describes Gopakumar--Vafa (GV) invariants associated to $cA_n$ singularities. We (1) generalise GV invariants to crepant partial resolutions of $cA_n$ singularities, (2) show that generalised GV invariants also satisfy Toda's formula and are determined by their associated contraction algebra, (3) give filtration structures on the parameter space of contraction algebras associated to $cA_n$ crepant resolutions with respect to generalised GV invariants, and (4) numerically constrain the possible tuples of GV invariants that can arise. We further give all the tuples that arise from GV invariants of $cA_2$ crepant resolutions.
Henry Fontana (UIC): The Harder-Narasimhan Filtration of a Trigonal Canonical Curve
A trigonal canonical curve C lies on a surface scroll in P^{g-1}. We use this fact to compute the Harder-Narasimhan filtration for the normal bundle of a general such C.
Jon Kim (CU Boulder): Moduli of (b, c)-Weighted Stable Marked Cubic Surfaces
In 2024, Nolan Shock constructed several KSBA compactifications of the moduli space of cubic surfaces. More precisely, he considered pairs consisting of a cubic surface and a boundary divisor given by the sum of the 27 lines, all with the same weight value in the interval (1/9, 1] and provided a finite wall-and-chamber decomposition. and described the weighted stable pairs parameterized by the moduli spaces in each chamber. In this poster, I will describe recent work where I provide a similar finite wall-and-chamber decomposition for KSBA compactifications where we weight one “heavy” line with weight b, and the other 26 lines uniformly with weight c.
Martina Miseri (Roma 3): The Prym-canonical Clifford index
Given a smooth curve C, we define a new Clifford index computed with respect to the Prym-canonical bundle, that is \omega_C \otimes \eta, where \eta is a 2-torsion nontrivial line bundle. Inspired by the theory about the classic Clifford index, we state a Clifford's Theorem that guarantees the non negativity of this new index and describes the curve when it is zero. We classify curves with low Prym-canonical Clifford index and we compute it for general curves. Finally, we consider the case of hyperelliptic curves where the Prym-canonical Clifford index keeps track of the geometry of the curve, as it depends on the nontrivial 2-torsion line bundle chosen on C.
Riku Kurama (Michigan): Flops, derived categories and liftability of threefolds
We present on the joint work with Alexander Perry (in preparation at the time of submitting this abstract) where we use derived category techniques to prove birational invariance of liftability for certain threefolds in positive characteristic. Roughly stated, a recent work by Hacon and Witaszek proved among other results that a (strict) Calabi-Yau threefold in positive characteristic birational to a liftable (strict) Calabi-Yau threefold is again liftable, assuming the existence of certain resolutions of singularities in mixed characteristic. Their proof uses the minimal model program in mixed characteristic, but our work approaches this result from a different direction by extending Bridgeland’s interpretation of threefold flops via derived categories.
Ronan O'Gorman (UC Berkeley): Group chunks and configurations
In 2012 it was shown by Zilber, using model theory, that a smooth curve of genus >1 over an algebraically closed field k can be reconstructed from the group structure on the k-points of its Jacobian, together with the k-points of the image of the curve under an embedding into the Jacobian. We will present ongoing work on how this proof can be understood and generalized using algebro-geometric methods, and in particular we will discuss how to construct group varieties from families of birational morphisms. This generalizes work of Weil and Artin constructing a group from a rationally defined group law.
Saket Shah (Michigan): Derived categories and flips for del Pezzo varieties
For a general del Pezzo variety of degree 3 or 4 (i.e. a cubic hypersurface or a complete intersection of two quadrics), we construct a standard flip between the Fano variety of k-dimensional quadrics on X and another variety, depending on the degree of X. We explore also some derived categorical consequences of this fact.
Sixuan Lou (UIC): Threefolds containing all curves are rationally connected
Any smooth projective curve embeds into P^3. More generally, any curve embeds into a rationally connected variety of dimension at least three. We prove conversely that if every curve embeds in a threefold X, then X is rationally connected. In particular "all curves embed" is a birational property for threefolds.
Amal Mattoo (Columbia): Objects of a Phantom on a Rational Surface
In 2023, Johannes Krah discovered a phantom on a rational surface, i.e., an admissible subcategory of the derived category with vanishing Grothendieck group. We construct several families of objects in this phantom and show results recovering some of the geometry of the surface from the phantom. We also construct a generator of the phantom whose endomorphism algebra lies in non-negative degrees, answering a question of Ben Antieau.
Andreas Kretschmer (HU Berlin): Some Characteristic Numbers for Cubic Hypersurfaces
Building on Aluffi's space of complete plane cubic curves, we generalize his construction to higher dimensions, resulting in a so-called 1-complete variety of cubic hypersurfaces. This solves the excess intersection problem of computing the number of smooth cubic hypersurfaces passing through m general points and tangent to n general lines, in principle in any dimension. A blow-down of the space of complete quadrics enters the picture at a crucial point of the construction. From this one may also compute characteristic numbers with respect to points and lines for other families of cubic hypersurfaces, e.g. singular ones, or those containing a given line. This is currently being done in a follow-up project.
Giusi Capobianco (Roma 2): The moduli space of double covers of hyperelliptic curves of genus g and its tropical counterpart
Given a smooth hyperelliptic curve C of genus g it is possible to construct all the distinct étale double covers of C by subdividing the branch locus B of the hyperelliptic map into two distinct even and non-empty subsets B1 and B2. Construct two hyperelliptic curves C1 and C2 branched over B1 and B2 respectively and the fibred product of C1 and C2 over \mathbb P^1 is exactly the source curve of the cover. We prove that a similar construction can be developed in the tropical setting for free double covers of hyperelliptic graphs. Moreover, the source curve/graph is not necessarily hyperelliptic and we recall the notion of h-hyperellipticity in order to describe the locus of the double covers of curves/graphs inside the moduli space R_g of double covers of curves (resp tropical curves).
Crislaine Kuster (IMPA): Codimension one foliations on adjoint varieties
Let X be a homogeneous variety, i.e., a variety that admits a Lie group acting on it transitively. In this poster, I will explore the space of codimension one foliations on X with a fixed normal sheaf N, focusing on the simplest possible choice of N. In particular, considering X an adjoint variety – a specific type of homogeneous variety – embedded in a projective space, we will be interested in determining whether a foliation on X is a restriction of a foliation on the ambient space P^n.
Daksh Aggarwal (Brown): Brill-Noether theory for totally ramified covers of P^1
We study the Brill-Noether theory of covers of the projective line totally ramified at two points and present progress towards conjectures of Pflueger on this problem using the techniques of degeneration and limit linear series.
Enhao Feng (Boston College): Moduli space of genus one curves on smooth cubic threefold
Let X be a smooth cubic threefold. By invoking ideas from the Geometric Manin's conjecture, we classify the main components of the Kontsevich moduli space of genus one stable maps on X. In particular, we show that when the degree of the map is at least five, there are exactly two irreducible components, of which one generically parametrizes maps that are birational onto their images, and the other parametrizes maps that are coverings of lines.
Ethan Partida (Brown): Graded Ehrhart Theory of Unimodular Zonotopes
Ehrhart theory studies the lattice point counts of integer dilations of lattice polytopes. Recently, Reiner and Rhoades introduced a notion of graded Ehrhart theory and conjectured that it satisfies many nice properties. We connect the graded Ehrhart theory of unimodular zonotopes to Tutte polynomials of matroids and certain line bundles on matroid Schubert varieties. We use these connections to prove the conjectures of Reiner and Rhoades in the case of unimodular zonotopes. This is one of the first large family of polytopes where the Reiner and Rhoades conjectures are known to hold.
Fernando Figueroa (Northwestern): Log Calabi-Yau pairs of complexity zero and arbitrary index
Log Calabi--Yau Pairs are a generalization of Calabi--Yau varieties, naturally occurring when considering families or branched covers. The Complexity of a Calabi--Yau pair measures how far it is from being a toric pair. More concretely, Brown, McKernan, Svaldi and Zong proved that any Calabi-Yau pair of index one and complexity 0 is a toric pair. Recent work of Mauri and Moraga has studied its crepant birational analogue, the "birational complexity", which measures how far the pair is from admitting a birational toric model. In this poster we predent some extensions of the previously known results for Calabi--Yau pairs of index one to arbitrary index. In particular we completely characterize Calabi--Yau pairs of complexity zero and arbitrary index. This is based on joint work with Joshua Enwright.
Gleb Terentiuk (Michigan): Ogus-Vologodsky equivalence via stacks
Ogus-Vologodsky equivalence can be seen as a characteristic p analogue of nonabelian Hodge theory. It turns out this equivalence can be seen using the theory of prismatization developed by Drinfeld and Bhatt-Lurie and the goal of my poster is to explain this approach.
Haoming Ning (Washington): Higher Du Bois and Higher Rational Pairs
Du Bois and rational singularities are some of the most important singularities studied in algebraic geometry due to their nice cohomological behavior. Recently, driven by developments in Hodge theoretic methods, there has been substantial interest in studying their higher analogs. We follow a principle of the minimal model program — that one should always study singularities via pairs — and propose a framework for higher Du Bois and rational pairs in the general setting. We extend numerous results to these higher pairs, including m-rational implies m-Du Bois, Bertini type theorems and a Kovács—Schwede type injectivity theorem. This is based on joint work with Brian Nugent.
Hyunsuk Kim (Michigan): Local cohomology and singular cohomology of Toric Varieties
Given an affine toric variety $X$ embedded in a smooth variety, we prove a general result about the mixed Hodge module structure on the local cohomology sheaves of $X$. As a consequence, we prove that the singular cohomology of a proper toric variety is mixed of Hodge--Tate type. Additionally, using these Hodge module techniques, we derive a purely combinatorial result on rational polyhedral cones that has consequences regarding the depth of reflexive differentials on a toric variety. We then study in detail two important subclasses of toric varieties: those corresponding to cones over simplicial polytopes and those corresponding to cones over simple polytopes. Here, we give a comprehensive description of the local cohomology in terms of the combinatorics of the associated cones, and calculate the Betti numbers (or more precisely, the Hodge--Du Bois diamond) of a projective toric variety associated to a simple polytope.
Marta Benozzo (LMO Paris-Saclay): Anti-Iitaka inequality in positive characteristic
An important invariant to classify varieties is their Kodaira dimension, which measure the positivity of the canonical divisor. Over the complex numbers, Iitaka conjectured that Kodaira dimensions in fibrations satisfy subadditivity. Instead, for the Iitaka dimensions of the anticanonical divisors, a superadditivity inequality holds for fibrations over a field of any characteristic, provided that the singularities of the anticanonical Q-linear system of the ambient space behave “nicely” on the general fibre. In characteristic 0, it is enough to control the “geometry” of these singularities, whereas in positive characteristic we need to control also their “arithmetic properties”. More precisely, in this latter case, we ask that some spaces that measure F-splitting properties are trivial along the general fibre. This is joint work with Brivio and Chang.
Morena Porzio (Columbia): On the Stable Birationality of Hilbert schemes of points on surfaces
In this work, I have addressed the question for which pairs of integers (n,n') the variety Hilb^n_X is stably birational to Hilb^n'_X, when X is a surface with H^1(X, O_X)=0. In order to do so we relate the existence of degree n' effective cycles on X with the existence of degree n ones using curves on X. One of the results is that, for geometrically rational surfaces, there are only finitely many stable birational classes among the Hilb^n_X 's. As a corollary, the rationality of a generalization of the Hasse-Weil zeta function Z(X, t) in K_0(Var/k)/([A^1_k])[[t]] (in char(k) = 0) is deduced.
Roktim Mascharak (Imperial): Birational Geometry of Three-folds versus Birational Geometry of Foliations on Three-fold
The Minimal Model Program (MMP) is a major tool to understand Birational Geometry of algebraic varieties and the Sarkisov Program is a byproduct of MMP which relates different Mori fiber spaces in same birational class (roughly). We will try to explore how similar (or different) the Birational Geometry of foliated three-folds are, to that of the usual algebraic varieties and how does the geometry depend on the rank of the foliation.
Rose Lopez (UC Berkeley): Gerbe Structures of Moduli Spaces
Since every elliptic curve has an involution, the moduli stack of elliptic curves is a \mu_2-gerbe over its rigidification, and we can ask what the class of the gerbe is, as an H^2 cohomology class. Similarly, since every genus 2 curve is hyperelliptic, the moduli stack of genus 2 curves is a \mu_2-gerbe over its rigidification, and we can ask for its H^2 cohomology class. More generally, we can consider the locus of curves in M_g which have an order N automorphism. This stack is a \mu_N-gerbe over its rigidification. We study the case where the quotient of the curve by an order N automorphism has genus 0 and determine the cohomological Brauer class of such loci.
Siao Chi Mok (Cambridge): Logarithmic Fulton—MacPherson configuration spaces
The Fulton—MacPherson configuration space is a well-known compactification of the ordered configuration space of a projective variety. We extend the construction to the logarithmic setting: it is a compactification of the configuration space of points on a projective variety X away from a simple normal crossings divisor D, and is constructed via logarithmic geometry. Moreover, given a simple normal crossings degeneration of X, logarithmic geometry similarly enables a logarithmically smooth degeneration of the Fulton—MacPherson space of X. Both constructions parametrise point configurations on certain target degenerations arising from both logarithmic geometry and the original Fulton–MacPherson construction. The degeneration satisfies a “degeneration formula” – each irreducible component of its special fibre can be described as a proper birational modification of a product of logarithmic Fulton–MacPherson configuration spaces.
Sridhar Venkatesh (Michigan): Local cohomological dimension of toric varieties
We relate the local cohomological dimension (lcd) of an affine toric variety with the Lefschetz morphism on the singular cohomology of a projective toric variety of one dimension lower. As a corollary, we show that the lcd is not a combinatorial invariant. We also provide a recipe for producing examples of toric varieties in every dimension with any possible lcd. This is based on joint work with Hyunsuk Kim.
Joshua Enwright (UCLA): Complexity one varieties are cluster type
The complexity of a pair (X,B) is an invariant that relates the dimension of X, the rank of the group of divisors, and the coefficients of B. If the complexity is less than one, then X is a toric variety. We prove that if the complexity is less than two, then X is a Fano type variety. Furthermore, if the complexity is less than 3/2, then X admits a Calabi–Yau structure(X,B) of complexity one and index at most two, and it admits a finite cover Y→X of degree at most 2, where Y is a cluster type variety. In particular, if the complexity is one and the index is one, (X,B) is cluster type. Finally, we establish a connection with the theory of 𝕋-varieties. We prove that a variety of 𝕋-complexity one admits a similar finite cover from a cluster type variety. This is joint work with Jennifer Li and José Yáñez.