Algebraic Geometry Seminar
Fall 2025
Regular Day: Tuesday
Regular Time: 3:30PM - 4:30PM
Regular Location: LCB 323
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August 19 |
José Ignacio Yáñez
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The complexity of a Calabi-Yau pair (X,B) is an invariant that relates the dimension of X, the Picard rank of X, and the coefficients of B. It was proven by Brown, McKernan, Svaldi and Zong that the complexity of a Calabi-Yau pair is nonnegative, and a variety X admits a Calabi-Yau pair of complexity 0 if and only if X is toric. In this talk we will discuss the geometry of Calabi-Yau pairs of index one and complexity one or two. Both descriptions are done in terms of cluster type varieties, a generalization of toric varieties. In the case of complexity one, we prove that (X,B) is of cluster type. In the case of complexity two, we give a criterion to decide whether the pair is cluster type or not. This is joint work with Joshua Enwright, Jennifer Li and Joaquin Moraga.
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August 27 Wednesday @4pm in LCB 222 |
Chenyang Xu
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(Joint with Ziquan Zhuang) In this lecture, I will explain our boundedness results for klt singularities with normalized volume bounded from below by a positive constant.
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September 2 |
TBA
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September 9 |
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September 16 |
Daigo Ito
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In the study of derived categories of coherent sheaves, ample line bundles play a fundamental role -- their tensor powers generate the derived category. This raises a natural question: does this generation property characterize ampleness? The answer is negative, but we show that this categorical property can be checked by a classical numerical criterion naturally extending the Nakai--Moishezon criterion. Moreover, the cone of divisors satisfying this condition lies between the big cone and the ample cone. In this talk I will focus on explaining the case of surfaces, where the geometry becomes especially clear. This is a joint work with Noah Olander.
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September 23 |
Yotam Svoray
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The ADE classification is an important collection of objects that appears in many fields of mathematics. In the context of algebraic geometry and commutative algebra, the ADE classification proved an important collection of hypersurfaces with unique properties, that were classified by Artin, Du Val, Arnold, Greuel, and many others. In this talk we will discuss the different properties of the ADE classification over fields of both zero and positive characteristic, and how we can generalize these results to over complete regular local rings using a characteristic free approach.
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September 30 |
Phil Tosteson
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For a projective variety X defined over a finite field, the homology of the space of rational curves on X is closely related to the F_q(T) rational points on X. In particular, Manin's conjecture for the asymptotic count of rational points on a Fano variety suggests a corresponding homological stability conjecture for moduli spaces of rational curves. We will discuss joint work with R. Das, B. Lehmann, and S. Tanimoto establishing both these conjectures in the case where X is a quartic del Pezzo surface.
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October 7 |
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October 14 @4pm |
Brian Nugent
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Rational and Du Bois singularities are among the most important classes of singularities in algebraic geometry because of their nice homological properties and their relation to the singularities of the minimal model program. Recently, there has been a lot of progress in studying their higher analogs. I will talk about recent work (joint with Haoming Ning) on generalizing the notion of higher Du Bois singularities to pairs.
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October 21 |
TBA
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October 28 |
Lisa Marquand
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Cubic fourfolds have been classically studied up to birational equivalence, with an eye towards rationality problems. We will discuss two other notions of equivalence: Fourier-Mukai equivalence, and `hyperkahler equivalence'. We'll discuss how these equivalences are conjecturely related. We will give new examples of pairs of cubic fourfolds satisfying all three equivalences. In particular, our examples will be pairs of birational cubic fourfolds, with birational Fano varieties of lines, a previously unknown phenomenon. This is joint work with Corey Brooke and Sarah Frei.
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November 4 |
Lingyao Xie
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For a projective morphism f: X to S, we explore when it is possible to extend a divisor that is numerically trivial over an open subset to a global relatively num-trivial divisor. In particular, we show that such $L$ always exists after a (weak) semi-stable reduction when $\dim S=1$.
On the other hand, we give an example showing that $L$ may not exist (after any reasonable modification of $f$) if $\dim S\ge 2$, which also gives an $f_U$-nef divisor $M_U$ that cannot extend to an $f$-nef ($\mathbb{Q}$) divisor $M$ for any compactification of $f|_U$, even after replacing $X_U$ with any higher birational model.
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November 11 |
Zach Mere
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Kontsevich introduced motivic integration to prove that birationally equivalent complex Calabi-Yau manifolds have the same Hodge numbers. The proof uses a change-of-variables formula which assumes the varieties involved are smooth. In this talk, we'll discuss a generalization of this formula to the case where singularities are allowed. Then we'll deduce a result about Calabi-Yau varieties over an arbitrary field.
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November 18 |
Shikha Bhutani
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Vanishing theorems are foundational tools in the Log Minimal Model Program, with the Kawamata-Viehweg Vanishing Theorem being one of the most important. However, these fail in positive characteristics. Fano varieties and their log generalizations are expected to behave better in this setting. In this talk, we prove the Kawamata--Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic p>5 and discuss consequences to threefold singularities.
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November 25 |
Riku Kurama
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