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Program

Higher Dimensional Algebraic Geometry: James 60 is a conference highlighting recent progress in higher-dimensional algebraic geometry scheduled in honor of James McKernan's 60th birthday.  The conference will focus on the many areas of algebraic geometry impacted by the groundbreaking work of James and his collaborators.

The photo of Ellen Browning Scripps Memorial Pier was taken by David deLeon and is used under an Unsplash license.  The photo of James McKernan was taken by Brigitte Lacombe and used on the Breakthrough Prize website.  The photo of the Insitute of the Americas was taken from the UCSD library website.

List of speakers:

Adebisi Agboola (UC Santa Barbara)

 

Valery Alexeev (University of Georgia)

Florin Ambro (Simion Stoilow Institute)

Lucia Caporaso (Roma Tre University)

 

Alessio Corti (Imperial College London and WUSTL)

 

Kristin DeVleming (UMass Amherst)

Christopher Hacon (University of Utah)

 

Elham Izadi (UC San Diego)

Anne-Sophie Kaloghiros (Brunel University)

Yujiro Kawamata (University of Tokyo)

 

Sándor Kovács (University of Washington)

 

John Lesieutre (Penn State University)

Michael McQuillan (University of Rome Tor Vergata)

Mircea Mustață (University of Michigan)

 

Rahul Pandharipande (ETH Zürich)

 

Nicholas Shepherd-Barron (King's College London)

Vyacheslav Shokurov (Johns Hopkins University)

Calum Spicer (King's College London)

Roberto Svaldi (University of Milan)

Burt Totaro (UC Los Angeles)

Jarosław Włodarczyk (Purdue University)

Registration

We ask all participants (including speakers) to fill out this registration form.  Funding decisions will be sent out in early November.  Please contact the organizers with any questions.

 

Funding for the conference has generously been provided

by the National Science Foundation under Grant DMS-2327037.

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Schedule

All talks will be held at the Institute of the Americas, Hojel Auditorium (Copley International Conference Center) on the main campus of UCSD.  Directions are below.

There are links to talk slides in the Abstracts section.

Every morning there will be coffee and pastries starting at 9 am.  We ask all participants to check-in during one of the registration times.

On Thursday there will be a poster session following the talks.  Please participate and support our early-career mathematicians!

Directions

Directions to Hojel Auditorium: The Institute of the Americas Complex is composed of three Spanish colonial style buildings. The Hojel Auditorium is one of these three buildings. Here is the Google Maps link.

Walking directions: If you prefer to have a direct link for walking directions from various spots on campus, you can refer to this pdf.

Public transportation directions: More information about public transportation options can be found here.

Rideshare directions: The address for rideshare is 9899 International Ln, La Jolla, CA 92093.

Driving directions: The closest parking to the Hojel Auditorium is the Pangea parking structure. Please park in Visitor (V) spots. Payment can be done by credit card or via the app ParkMobile. Please allow 15 minutes to locate parking, buy a parking permit and walk to the complex. More information can be found here.

Abstracts

Some of the titles will link to the slides used during the conference.

Adebisi Agboola (UC Santa Barbara)

Line bundles on abelian schemes and p-adic heights

Let F be a number field with ring of integers O_F, and suppose that E/O_F is an abelian scheme. If p is a prime of ordinary reduction of E, to what extent is an element of Pic^{0}(E) determined by its restriction to p-power torsion subgroup schemes of E?

I shall discuss an answer to this question that involves a new construction of the p-adic height pairing associated to E. This is joint work in progress with F. Castella and M. Ciperiani.

Valery Alexeev (University of Georgia)

Compact moduli of K3 and Enriques surfaces

I will report on recent advances in understanding geometric, KSBA stable pair compactifications of K3 and Enriques surfaces.

Florin Ambro (Simion Stoilow Institute)

On toric Fano fibrations

We discuss the classification of germs of toric Fano fibrations, extending work of A. Borisov in the case of Q-factorial toric singularities. As an application, we verify in the toric case a conjecture of V. Shokurov on the existence of complements with bounded index and prescribed singularities.

Lucia Caporaso (Roma Tre University)

Hypertangency of plane curves and algebraic hyperbolicity

The talk will be about joint work in progress with Amos Turchet, investigating pairs of plane curves intersecting in at most two points.

The principal motivation comes from a series of conjectures of Vojta and others on the hyperbolicity of the complement (in the projective plane) of a curve of degree at least 4 with normal crossing singularities. We apply our results to some cases of these conjectures.

Alessio Corti (Imperial College London and Washington University in St. Louis)

Smoothing admissible toric Fano 3-folds

I will describe a sufficient criterion for smoothing a toric Fano 3-fold. I will also discuss how the criterion fits in the Fano/LG correspondence. This is work with Paul Hacking and Andrea Petracci.

Kristin DeVleming (UMass Amherst)

K-moduli of a family of conic bundle threefolds

Recently, there has been significant progress in understanding the K-moduli spaces of Fano varieties and log Fano pairs (X,cD). When D is a rational multiple of the anticanonical divisor of X, the K-moduli spaces of log Fano pairs (X,cD) admit a wall crossing framework as c varies and there is a finite collection of rational values of c where the K-moduli spaces change. With Lena Ji, Patrick Kennedy-Hunt, and Ming Hao Quek, we explore the K-moduli spaces in an example where D is not proportional to the anticanonical divisor. We study the K-moduli space of pairs (P1xP2, cD) where D is a (2,2) divisor and prove that there is exactly one irrational value of c where the moduli spaces change. We further relate these moduli spaces to several related spaces: the GIT of (2,2) divisors in P1xP2, K-moduli of the conic bundle threefold that is the double cover of P1xP2 branched over D, and various moduli spaces of quartic plane curves arising as the discriminant of these conic bundles.

Christopher Hacon (University of Utah)

The work of James McKernan

I will survey some of James' most important contributions to algebraic geometry.

Elham Izadi (UC San Diego)

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.

Anne-Sophie Kaloghiros (Brunel University)

On K-moduli spaces of Fano 3-folds

In this talk, I will discuss the components of K-moduli spaces associated to some of the 105 deformation families of smooth Fano 3-folds. I will discuss all 1-dimensional components of K-moduli spaces of Fano 3-folds and a 3-dimensional component parametrising K-polystable Fano 3-folds with a smoothing which is a (1,1,1,1) divisor in (P^1)^4.

 

This talk is based on joint projects with Abban, Cheltsov, Denisova, Etxabarri-Alberdi, Jiao, Martinez-Garcia and Papazachariou, and with Cheltsov, Fedorchuk and Fujita.

Yujiro Kawamata (University of Tokyo)

On derived McKay correspondence between non-commutative deformations of commutative and non-commutative minimal resolutions

We consider an example of derived McKay correspondence between non-commutative (NC) deformations in the case of surface singularities of type An. We construct a versal NC deformation of the commutative crepant resolution and compare it with the versal NC deformation of the non-commutative crepant resolution. We show the derived McKay correspondence in the case n=1.

Sándor Kovács (University of Washington)

On higher Du Bois singularities

Du Bois singularities were first defined and studied by Steenbrink, motivated by Hodge theoretic interests. Soon after it was recognized by Kollár that, due to their good deformation theoretic properties, Du Bois singularities play an important role in the moduli theory of higher dimensional varieties. Recently, returning to the initial Hodge theoretic motivation, Friedman and Laza and a bit later Mustață and Popa started the systematic study of higher degree analogues of Du Bois singularities.
In this talk, I will discuss a generalization of a result of Mustață and Popa as well as the deformation invariance of higher Du Bois singularities. 

 

John Lesieutre (Penn State University)

More pathologies of the volume function

I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudoeffective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.

Michael McQuillan (University of Rome Tor Vergata)

Thurston Vanishing

Thurston's method of proof in establishing that rational maps $f$ of $\mathbb{P}^1_{\mathbb{C}}$ with pre-periodic critical points which aren't multiplication on an elliptic curve $\pm 1$ are rigid was generalised by Adam Epstein to an almost exhaustive local to global theory of deformations of dynamical systems. Indeed arguably its only fault was not to have identified a suitable topus of $f$-equivariant sheaves. Once one remedies this lacuna, natural extensions of Epstein's methods and new results in dynamical systems immediately suggest themselves. En passant we calculate the dualising complex of a real blow up, which is not only a sheaf but a remarkably algebraic one which presents notable opportunities for trivialising the analysis of Stokes' phenomenon. A pre-print is available at, https://arxiv.org/abs/2311.02518.

Mircea Mustață (University of Michigan)

A refined log canonical threshold for local complete intersections

The minimal exponent of a hypersurface is a refinement of the log canonical threshold that gives a numerical measure of rational singularities. I this talk I will discuss an extension of the minimal exponent to local complete intersections and explain how it can be used to characterize the higher order version of Du Bois and rational singularities in this setting. This is based on joint work with Qianyu Chen, Brad Dirks, and Sebastian Olano.

Rahul Pandharipande (ETH Zürich)

Cycles on the moduli space of abelian varieties

I will present results and conjectures related to algebraic cycles on the moduli space of principally polarized abelian varieties (with connections to parallel questions on the moduli spaces of curves and K3 surfaces).  Joint work with S. Canning, 

S. Molcho, and D. Oprea.

Nicholas Shepherd-Barron (King's College London)

The cohomology of elliptic surfaces

A variational approach shows that the (1,1) part of the primitive cohomology of an elliptic surface has a natural orthonormal basis. This leads to a new kind of period matrix for these surfaces.

Vyacheslav Shokurov (Johns Hopkins University)

Simple Fano varieties

Calum Spicer (King's College London)

Flips and Flops and Foliations

Due to recent progress in the birational geometry of foliations, it is known that minimal models exist for foliations in a wide range of cases. The key technical ingredient in showing the existence of minimal models in these cases is showing the existence of flips.  In this talk we will consider a related surgery operation, the flop.

In contrast to the case of varieties, flops do not exist in general, however, we are able to demonstrate the existence of flops for co-rank one foliations.  We will explain a proof of the existence of flops in this case, with a focus on the interplay between the study of singularities from the perspective of the MMP and the classical study of foliation singularities going back to Poincare and Dulac.  Features joint work in progress with P. Cascini.

Roberto Svaldi (University of Milan)

The birational structure of log Calabi-Yau pairs

In the first part of the talk I will discuss recent progress concerning our understanding of the birational structure of log Calabi-Yau pairs that was influenced and inspired by results/conjectures/idea of James McKernan and his collaborators.
In the second part, I will explain the solution to a conjecture of Shokurov characterizing toric varieties and toric singularities (which will still have to do with log Calabi-Yau pairs). This part will feature joint work with Brown, McKernan, Zong, and with Moraga.

Burt Totaro (UC Los Angeles)

Some failures of vanishing theorems

The minimal model program for 3-folds has been developed only in characteristics p at least 5. A key difficulty at small primes is that the singularities occurring in the minimal model program need not be Cohen-Macaulay, as they are in characteristic zero. This is related to the fact that Kodaira vanishing fails even on Fano varieties.  We present the latest work in this direction.

Jarosław Włodarczyk (Purdue University)

Resolution of Singularities of Varieties with Foliations by Torus Actions (jointly with D.Abramovich, A.Belotto and M.Temkin)

We discuss the resolution method by the operation of cobordant blow-ups as introduced in my paper “Resolution by torus actions” and its generalization in “Cox rings of morphisms and Resolution of singularities”. A similar notion was considered independently by Rydh in the stack-theoretic context.

 

The papers and the methods are based upon the ideas of the joint work with Abramovich and Temkin and a similar result by McQuillan on resolution in characteristic zero via stack-theoretic weighted blow-ups, and the paper of Abramovich-Quek on multiple weighted blow-ups.

 

The method gives the straightforward and efficient embedded resolution of varieties and the principalization of ideals, expressed in the language of torus actions and Rees algebras on smooth ambient varieties with simple normal crossings (SNC) divisors.

 

In the most recent paper written jointly with Abramovich, Belotto, and Temkin the method is applied in the context of foliated varieties leading, in particular, to the desingularization of the cross sections.

Posters

Iacopo Brivio (Center for Mathematical Sciences and Applications)

Superadditivity of anticanonical Iitaka dimension for fibrations in characteristic p>0

Let $f: X \to Y$ be a fibration of smooth projective complex manifolds, with general fiber $F$. When the anticanonical Q-linear system of $X$ has good singularities, a recent result of Chang established the Iitaka-type inequality $\kappa(-K_X) \leq \kappa(-K_F) + \kappa(-K_Y)$. Although such an inequality is known to fail in positive characteristic, we are able to extend this result to a large class of tame fibrations. As a byproduct we obtain a Bertini-type theorem for semiample anticanonical graded linear series on globally F-split varieties. This is based on joint work with Marta Benozzo and Chi-Kang Chang.

Benjamin Church (Stanford)

The fibering genus of Fano hypersurfaces

The "fibering genus" of X is the minimal genus of a rational fibration on X by curves. The fibering genus of a variety has been studied in work of Konno, Ein--Lazarsfeld, and Voisin in the non-Fano range. However, this measure of irrationality, unlike covering gonality or covering genus, can be interesting even for rationally-connected varieties. We prove lower bounds for the fibering genus of very general Fano hypersurfaces. In particular, we exhibit Fano varieties with arbitrarily large fibering genus. As an application, we rule out Mori fiber space structures of low relative dimension on very general Fano hypersurfaces giving a weak analog of "birational superrigidity" for higher index. The method follows Kollár's technique of degeneration to p-cyclic covers in characteristic p. The main difficulty is to ensure the degenerated fibration in curves over characteristic p is generically smooth for which a crucial import is Tate's genus change formula.

 

 

Louis Esser (Princeton)

The Dual Complex of a G-variety

We introduce a new invariant of G-varieties, the dual complex, which roughly measures how divisors in the complement of the free locus intersect. We show that the top homology group of this complex is an equivariant birational invariant of G-varieties. As an application, we demonstrate the non-linearizability of certain large abelian group actions on smooth hypersurfaces in projective space of any dimension and degree at least 3.

 

Fernando Figueroa (Princeton)

Fundamental groups of low coregularity Calabi-Yau type pairs

It was shown by Braun that the fundamental group of the smooth locus of Fano type pairs is finite. This is false for Calabi-Yau type pairs, even in dimension one. The purpose of this poster is to present how the fundamental group behaves in low coregularity Calabi-Yau type pairs.  The main theorems are the finiteness of the fundamental group in coregularity 0 and the similarities between low coregularity and low dimension.
This is based on joint work with Lukas Braun.

Changho Han (University of Waterloo)

Extending Torelli map from the moduli space of fold-like curves

It is well-known that the Torelli map, that turns a smooth curve of genus g into its Jacobian (a principally polarized abelian variety of dimension g), extends to a map from the Deligne—Mumford moduli of stable curves to the moduli of semi-abelic varieties by Alexeev. Moreover, it is also known that the Torelli map does not extend over the alternative compactifications of the moduli of curves as described by the Hassett—Keel program, including the moduli of pseudostable curves (can have nodes and cusps but not elliptic tails). But it is not yet known whether the Torelli map extends over alternative compactifications of the moduli of curves described by Smyth; what about the moduli of curves of genus g with rational m-fold singularities, where m is a positive integer bounded above? As a joint work in progress with Jesse Kass and Matthew Satriano, I will describe moduli spaces of curves with m-fold singularities (with topological constraints) and describe how far the Torelli map extends over such spaces into the Alexeev compactifications.

Daigo Ito (UC Berkeley)

Gluing of Fourier-Mukai partners using tensor triangulated geometry

We initiate studies of the relationship between birational geometry of smooth varieties and tensor structures of their derived categories. A crucial insight from tensor triangulated geometry is that we can view a variety as a specific tensor structure on its derived category and an upshot is that closer observations on such tensor structures allow us to glue its Fourier-Mukai partners to construct a smooth scheme locally of finite type, which we call the Fourier-Mukai locus. The geometry of the locus reflects geometric properties of the Fourier-Mukai partners in various contexts, such as abelian, toric, and Fano varieties, as well as birational operations among them such as flops. We further investigate relations with the DK hypothesis and methods to purely categorically characterize the locus.

Jennifer Li (Princeton)

On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds

Let $Y$ be a smooth projective 3-fold admitting a $K3$ fibration $f: Y \rightarrow \mathbb{P}^{1}$ with $-K_{Y} = f^{\ast} \mathcal{O}(1)$. We show that the pseudoautomorphism group of $Y$ acts with finitely many orbits on the codimension one faces of the movable cone if $H^{3}(Y, \mathbb{C}) = 0$, confirming a special case of the Kawamata-Morrison-Totaro cone conjecture. In Coates-Corti-Galkin-Kasprzyk 2016, Przyjalkowski 2018, and Cheltsov-Przyjalkowski 2018, the authors construct log Calabi-Yau 3-folds with $K3$ fibrations satisfying the hypotheses of our theorem as the mirrors of Fano 3-folds.

Lisa Marquand (Courant Institute)

The defect of cubic threefolds

The defect of a cubic threefold X with isolated singularities is a global invariant that measures the failure of Q-factoriality. We compute the defect for such cubics by projecting from a singularity. We determine the Mixed Hodge structure on the middle cohomology of X in terms of both the defect and local invariants of the singularities. We then relate the defect to various geometric properties of X: in particular, we show that a cubic threefold is not Q-factorial if and only if it contains either a plane or a cubic scroll.

Lucas Mioranci (University of Illinois Chicago)

Algebraic hyperbolicity of very general hypersurfaces in homogeneous varieties

A complex projective variety $X$ is algebraically hyperbolic if there exists an ample divisor $H$ and a real number $\epsilon > 0$ such that the geometric genus $g(C)$ and the degree of any integral curve $C\subset X$ satisfy the inequality \[ 2g(C) - 2\ge \epsilon \deg_H (C). \]

 

The algebraic hyperbolicity is an important property to characterize varieties of general type, and it is connected to famous conjectures such as the Lang Conjectures and Green-Griffiths Conjecture.

 

By building on recent work, I classify algebraic hyperbolic hypersurfaces in homogeneous varieties, thus obtaining explicit bounds for the hyperbolicity in plenty of open cases, including Grassmannians, flag varieties, and their products.

Talon Stark (UCLA)

Partial positivity and asymptotic cohomology

Asymptotic cohomology generalizes the notion of volume for a line bundle. These invariants $\hat{h}^i(X,L) are defined exactly analogously to volume, with $h^i$ replacing $h^0$. In 2006, de Fernex, Kuronya, and Lazarsfeld proved that a line bundle $L$ is ample if and only if $L$ is asymptotically ample, ie. its asymptotic cohomology functions $\hat^i(X,-)$ vanish for all $i>0$ in a neighborhood of $L$ in $N^1(X)$. In 2010, Totaro asked whether this criterion extends to a criterion for q-ample line bundles. Explicitly, is a line bundle q-ample (satisfying Serre vanishing in degrees greater than $q$) if and only if it is asymptotically q-ample (defined by the local vanishing of $\hat{h}^i(X,-)$ in degrees $i>q$)? We give an affirmative answer to this question for line bundles $L$ such that $\dim B_{+}(L) \leq q+1$. This is based on joint work with Nathan Chen and Chengxi Wang. 

Fei Xiang (UC Irvine)

On the Crepant Resolutions and Quiver Moduli Spaces

It is first observed by McKay that there exists a one-to-one correspondence between the components of the exceptional locus of a resolution and the nontrivial irreducible representations of the group, which is known as McKay Correspondence. The existence of a crepant resolution and the equivalence between the derived categories have been proved in some particular cases.
This poster will be interested in the conjecture that the existence
of such a resolution, with some additional restriction, should result in one of the quiver moduli spaces. We will then give the main idea of the recovering process of the resolution from the derived category using Hochschild cohomology.

Wern Yeong (UCLA)

The algebraic Green-Griffiths-Lang conjecture for the complement of a generic quartic plane curve

A complex algebraic variety is said to be (Brody) hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. The Green-Griffiths-Lang (GGL) conjecture predicts that varieties of (log) general type are hyperbolic outside of a proper subvariety called an exceptional locus. We prove the algebraic version of this Conjecture (with respect to Demailly’s algebraic hyperbolicity) for the complement of a generic degree 2n hypersurface in Pn. Moreover, for the complement of a generic quartic plane curve, we completely characterize the exceptional locus as the union of the flex and bitangent lines. This is joint work with Xi Chen and Eric Riedl. 

Logistics

Lunch: Here are some lunch options close to the conference site, organized by distance:

Restaurants in the North Torrey Pines Living & Learning Neighborhood (8-10 min walk):​​

Restaurants in the Price Center:

A complete list of dining options on campus can be found here.

Hotels: UCSD is located in La Jolla, California.  The closest major airport is the San Diego International Airport.  The best way to get to UCSD from the airport is to use a taxi or ride-sharing service (such as Uber or Lyft).

There are several hotels in La Jolla, which may be a bit more expensive unless booked in advance. Current prices range from $205 to $240 per night; the first hotel may be able to offer UCSD rates, depending on availability.

Some other options close to the Empress are Inn by the Sea and La Jolla Village Lodge.  There are other hotels in the area that can be found on the standard websites.

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