Moduli of higher-dimensional varieties

Please find the tentative schedule below. Titles and abstracts for the afternoon talks can be found further down the page. The morning talks are plenary talks for all participants of all sessions and the afternoon talks listed are just those for our session. The afternoon talks on Monday will be held in the same location as the plenary talks, the Lory Student Center Grand Ballroom, and the afternoon talks on the other days will be held in Yates Hall Room 104.

Monday, July 14:

Morning (location: Lory Student Center Grand Ballroom):
9 - 9:50: Mattias Jonsson
10:30 - 11:20: Karl Schwede

Afternoon (location: Lory Student Center Grand Ballroom):
1:30 - 2:20: Ivan Cheltsov

2:50 - 3:40: Kristin DeVleming

4:10 - 5: Ziquan Zhuang

Tuesday, July 15:

Morning (location: Lory Student Center Grand Ballroom):
9 - 9:50: Sébastien Boucksom
10:30 - 11:20: Kevin Tucker

Afternoon (location: Yates Hall Room 104):
1:30 - 2:20: Anne-Sophie Kaloghiros

2:50 - 3:40: Yuchen Liu

4:10 - 5: Cinzia Casagrande

Wednesday, July 16:

Morning (location: Lory Student Center Grand Ballroom):
9 - 9:50: Chenyang Xu
10:30 - 11:20: Jakub Witaszek

Free Afternoon

Thursday, July 17:

Morning (location: Lory Student Center Grand Ballroom):
9 - 9:50: Chenyang Xu
10:30 - 11:20: Isabel Vogt

Afternoon (location: Yates Hall Room 104):
1:30 - 2:20: Hamid Abban

2:50 - 3:40: Lena Ji

4:10 - 5: Giancarlo Urzúa

Friday, July 14:

Morning (location: Lory Student Center Grand Ballroom):
9 - 9:50: Chenyang Xu
10:30 - 11:20: Zsolt Patakfalvi

Afternoon (location: Yates Hall Room 104):
1:30 - 2:20: Kenneth Ascher

2:50 - 3:40: Kento Fujita

4:10 - 5: Jarod Alper

Titles and abstracts

Hamid Abban
Title: Explicit K-stability of Fano varieties

Abstract: Foundations of K-stability for Fano varieties are well understood. Despite numerous methods in hand, explicit verification of K-stability remains a challenging problem in several directions. In this talk I will give an overview of the known results and some future directions.

Jarod Alper
Title: Moduli of Singular Curves and the Minimal Model Program of Mgbar

Abstract: We survey progress over the last decade on the construction and study of alternative compactifications of the moduli of smooth curves, with an emphasis on their relationship to the minimal model program. As a secondary goal, we use the moduli stack of all curves as a way to illustrate recent foundational developments on Theta-stability, good moduli spaces, and stratification

Kenneth Ascher
Title: The Hassett—Keel program in genus four

Abstract: The Hassett—Keel program seeks to give a modular interpretation to the log minimal model program for the moduli space of curves of genus g. The first few steps, in all genera, are known due to the work of Hassett—Hyeon and Alper—Fedorchuk—Smyth—van der Wyck. We will discuss the full picture in the case of genus four, supplementing previous results of Fedorchuk, Hyeon—Lee, and Casalaina-Martin—Jensen—Laza. Our main techniques are wall-crossing for moduli spaces in the sense of K-stability and KSBA-stability, and the recently constructed moduli space of boundary polarized Calabi-Yau surface pairs. This talk is based on joint work with Kristin DeVleming, Yuchen Liu, and Xiaowei Wang.

Cinzia Casagrande
Title: The Lefschetz defect and the classification of Fano 4-folds with large Picard number

Abstract: Let X be a smooth and complex Fano 4-fold, and rho(X) its Picard number. We will discuss the following theorem: if rho(X)>9, then X is a product of del Pezzo surfaces. This is sharp, as we know a family of Fano 4-folds with rho(X)=9 that is not a product of surfaces. A key ingredient in this result is the Lefschetz defect, an invariant that relates the Picard number of X to that of its prime divisors. We will discuss the properties of the Lefschetz defect of Fano varieties of arbitrary dimension, and then give an overview of its application to study of the geometry of Fano 4-folds with rho(X)>6, using birational geometry in the framework of MMP.

Ivan Cheltsov
Title: Low dimensional components in the K-moduli of smooth Fano 3-folds

Abstract: In this talk, I will survey what is known about irreducible components of the K-moduli of smooth Fano 3-folds that have small dimension (up to 3).

Kristin DeVleming
Title: Wall crossing for moduli spaces of varieties

Abstract: I will survey wall crossing results on moduli of varieties and pairs, focusing primarily on results for K-moduli of pairs (X,cD) as one allows the coefficient of D to vary such that (X,cD) is log Fano. I will then discuss several applications of the theory of wall crossing, including applications to moduli of K3 surfaces and moduli of Fano threefolds. This talk is based on several joint works with Kenneth Ascher and Yuchen Liu and work with Lena Ji, Patrick Kennedy-Hunt, and Ming Hao Quek.

Kento Fujita
Title: On the coupled Ding stability and the Yau-Tian-Donaldson correspondence for Fano manifolds

Abstract: We interpret the reduced coupled Ding stability of Fano manifolds in the notion of reduced coupled stability thresholds. As a corollary, we solve a modified version of the conjecture by Hultgren and Witt Nystroem for coupled Kaehler-Einstein metrics on Fano manifolds. This is a joint work with Yoshinori Hashimoto.

Lena Ji
Title: Rationality of some real conic bundle threefolds

Abstract: An algebraic variety is said to be rational if it is birational to projective space. In this talk, we discuss the rationality question over the real numbers for a certain class of conic bundle threefolds. The varieties we consider all become rational over the complex numbers, but in general the complex rationality construction need not descend to ℝ. We give a partial answer to the ℝ-rationality question, depending on the real topological type of the discriminant curve and its associated double cover. This talk is based on joint work with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt, and on joint work with Mattie Ji.

Anne-Sophie Kaloghiros
Title: K-polystable degenerations of prime Fano threefolds of genus 12.

Abstract: In the past couple of years, many components of K-moduli spaces parameterising K-polystable Fano threefolds with a smoothing in one of the 105 deformation families classified by Mori, Mukai and Iskovskikh have been explicitly described. In many cases, we now know which smooth Fano threefolds in the family are K-polystable and have some geometric information on K-(poly/ semi)stable degenerations. The case of prime Fano threefolds of genus 12 remains mysterious – we don’t even know which smooth prime Fano 3-folds are K-polystable (Donaldson’s conjecture). The associated component of the K-moduli space is 6-dimensional. In this talk, I will show that general one-nodal prime Fano threefolds of genus 12 are K-polystable. Prokhorov showed that there are 4 families of one-nodal prime Fano threefolds of genus 12 , and I will show that these 4 families correspond to 4 boundary components of the associated K-moduli. This is joint work with Elena Denisova.

Yuchen Liu
Title: Good moduli spaces for boundary polarized Calabi-Yau surface pairs

Abstract: While the theories of KSBA stability and K-stability have been successful in constructing compact moduli spaces of canonically polarized varieties and Fano varieties, respectively, the case of Calabi-Yau varieties (and pairs) remains less well understood despite many approaches in various cases. I will discuss a new approach to this problem in the case of boundary polarized Calabi-Yau pairs (X,D), i.e. X is a Fano variety and D is an anticanonical Q-divisor, in which we consider all semi-log-canonical degenerations. One challenge of this approach is that the moduli stack can be unbounded. Nevertheless, in dimensional two, we show that there exists a projective moduli space despite the unboundedness. As an application, we provide modular interpretation for Baily-Borel compactifications of moduli space of K3 surfaces with non-symplectic automorphisms. Based on joint work with Harold Blum that builds on previous work with Ascher, Bejleri, Blum, DeVleming, Inchiostro, and Wang.

Giancarlo Urzúa
Title: Wahl singularities in degenerations of del Pezzo surfaces

Abstract: After the work of Bădescu (1986), Manetti (1991), and Hacking (2001) on degenerations of rational surfaces, Hacking and Prokhorov (2010) classified all degenerations with only Wahl singularities of the complex projective plane. They correspond to partial Q-Gorenstein smoothings of weighted projective planes P(a^2,b^2,c^2), where (a,b,c) satisfies the Markov equation a^2+b^2+c^2=3abc. In a recent joint work with Juan Pablo Zúñiga (https://arxiv.org/abs/2504.19929), we classify all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree d, for any fixed 1<=d<=9. With that purpose in mind, we introduce del Pezzo Wahl chains with markings, which define marked del Pezzo surfaces. They control all such degenerations and are in one-to-one correspondence with particular fake weighted projective planes, just like in the case of degree 9. I plan to introduce marked Wahl chains and their del Pezzo surfaces, and sliding, which is the birational tool to understand the correspondence. I will mention some byproducts such as constraints on Wahl singularities for a given del Pezzo degree, and constructions of particular (Hacking's) exceptional collections of vector bundles, which gives a geometric proof of some recent results by Polishchuk and Rains.

Ziquan Zhuang
Title: Boundedness in general type MMP

Abstract: I will explain how local volumes (an invariant related to K-stability) can be used to show that in any general type MMP, the minimal log discrepancy of singularities takes only finitely many values, and the fibers of all flipping contractions and flips fall into finitely many deformation types. An application is the effective termination of fivefold general type MMP. This is based on joint work with Jingjun Han, Jihao Liu and Lu Qi.

Other week 1 sessions

Week 1 will run from July 14 - 18. The following information was taken from the main SRI website. Details, history, and an application to participate can be found here.

Morning Speakers

Sebastien Boucksom, Mattias Jonsson, Karl Schwede, Kevin Tucker, Isabel Vogt, Jakub Witaszek, Chenyang Xu, Susanna Zimmerman

Afternoon Sessions

1) Moduli of higher-dimensional varieties (organized by Carolina Araujo, Paolo Cascini, Kristin DeVleming)

2) Birational geometry in positive and mixed characteristics and connections to commutative algebra (organized by Linquan Ma, Zsolt Patakfalvi, Karen Smith)

3) Geometry of moduli (organized by Gavril Farkas, Sam Grushevsky, Isabel Vogt)

4) Analytic methods in algebraic geometry (organized by Philippe Eyssidieux, Mihai Paun, Christian Schnell)

5) Contributed talks

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