April 3: Daniel Litt (UGA)
Title: The section conjecture at the boundary of moduli space Abstract: Grothendieck's section conjecture predicts that over arithmetically interesting fields (e.g. number fields or p-adic fields), rational points on a smooth projective curve X of genus at least 2 can be detected via the arithmetic of the etale fundamental group of X. We construct infinitely many curves of each genus satisfying the section conjecture in interesting ways, building on work of Stix, Harari, and Szamuely. The main input to our result is an analysis of the degeneration of certain torsion cohomology classes on the moduli space of curves at various boundary components. This is (preliminary) joint work with Padmavathi Srinivasan, Wanlin Li, and Nick Salter. |
April 10: Ignacio Barros Reyes (Northeastern University)
Title: On product identities and the Chow rings of holomorphic symplectic varieties Abstract: For a moduli space M of stable sheaves over a K3 surface X, we propose a series of conjectural identities in the Chow rings CH_{*}(MxX^{l}), l≥1, generalizing the classic Beauville-Voisin identity for a K3 surface. We emphasize consequences of the conjecture for the structure of the tautological subring R_{*}(M)⊆ CH_{*}(M). We prove the proposed identities when M is the Hilbert scheme of points on a K3 surface. This is joint work with L. Flapan, A. Marian and R. Silversmith. |
April 17: Junliang Shen (MIT)
Title: Hitchin systems, hyper-Kaehler geometry, and the P=W conjecture Abstract: The P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In this talk, we will focus on interactions between compact and noncompact hyper-Kaehler geometries. Such connections, together with symmetries coming from the moduli of compact hyper-Kaehler manifolds, lead to new progress on the P=W conjecture. I will discuss the results we obtained and the difficulties we met. Based on joint work with Mark de Cataldo and Davesh Maulik. |
Special Enumerative Geometry Seminar
April 21: Andrea Ricolfi (SISSA) Title: Higher rank K-theoretic Donaldson-Thomas theory of points Abstract: Recently Okounkov proved Nekrasovs conjecture expressing the partition function of K-theoretic DT invariants of the Hilbert scheme of points Hilb(C^{3},points) on affine 3-space as an explicit plethystic exponential. We generalise Nekrasovs formula to higher rank, where the quot scheme of finite length quotients of O^{r} replaces Hilb(C^{3},points). This proves a conjecture of Awata-Kanno. Specialising to cohomological invariants, we obtain the statement of Szabos conjecture. We discuss some further applications if time permits. This is joint work with Nadir Fasola and Sergej Monavari. |
April 24: Gavril Farkas (Humbolt Universität, Berlin)
Title: Green's Conjecture via Koszul modules Abstract:Using ideas from geometric group theory we provide a novel approach to Green's Conjecture on syzygies of canonical curves. Via a strong vanishing result for Koszul modules we deduce that a general canonical curve of genus g satisfies Green's Conjecture when the characteristic is zero or at least (g+2)/2. Our results are new in positive characteristic (and answer positively a conjecture of Eisenbud and Schreyer), whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Joint work with Aprodu, Papadima, Raicu and Weyman. |
May 15: Pierrick Bousseau (ETH)
Note: due to time difference with Europe, the talk will be at 9am (pre-talk at 9am). Title: Quasimodular forms from Betti numbers Abstract: This talk will be about refined curve counting on local P^{2}, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of one-dimensional coherent sheaves on P^{2}. This gives a proof of some stringy predictions about the refined topological string theory of local P^{2} in the Nekrasov-Shatashvili limit. This work is in part joint with Honglu Fan, Shuai Guo, and Longting Wu |
May 22: Isabel Vogt (Stanford)
Title: Stability of normal bundles of space curves Abstract: In this talk we will introduce the basic notion of stability of vector bundles on curves and how it can be approached by degenerating the curve. We'll then apply this to the bundle that controls the deformation theory of a curve embedded in projective space: the normal bundle. This is joint work with Izzet Coskun and Eric Larson. |
May 29: Anand Deopurkar (ANU)
Title: Apparent boundaries of projective varieties Abstract: Fix a smooth projective variety in projective space and project it to a linear subspace of the same dimension. The ramification divisor of this projection is classically known as the "apparent boundary". How does the apparent boundary move when we move the center of projection? I will discuss the geometry arising from this natural question. I will explain why the situation is most interesting for varieties of minimal degree, and how it is related to limit linear series for vector bundles |
June 5: Behrouz Taji (University of Sydney)
Note: due to time difference with Australia, the talk will be at 4:30pm (no pre-talk). Title: Birational geometry of projective families of manifolds with good minimal models Abstract: A classical conjecture of Shafarevich, solved by Parshin and Arakelov, predicts that any smooth projective family of high genus curves over the complex line minus a point or an elliptic curve is isotrivial (has zero variation in its algebraic structure). A natural question then arises as to what other families of manifolds and base spaces might behave in a similar way. Kebekus and Kovács conjecture that families of manifolds with good minimal models form the most natural category where Shafarevich-type conjectures should hold. For example, analogous to the original setting of Shafarevich Conjecture, they expect that over a base space of Kodaira dimension zero such families are always (birationally) isotrivial. In this talk I will discuss a solution to Kebekus-Kovács Conjecture. |
June 12: Chenyang Xu (MIT)
Title: Algebraic K-stability theory of Fano varieties Abstract: In recent years, K-stability of Fano varieties has been proved to be a rich topic for higher dimensional geometers. The transition of knowledge is mutual. On one direction, we use the powerful machinery from higher dimensional geometry, especially the minimal model program, to have a better understanding of various concepts in K-stability. On the other direction, K-stability provides the right subclass to construct moduli spaces of Fano varieties, which had been once considered beyond reach by algebraic geometers. In the first half hour, I will explain how people change their viewpoint on the definition of K-stability. Then in the main talk, I will focus on the moduli of Fano varieties. |
Organizers: Elham Izadi, James McKernan and Dragos Oprea
This seminar is supported in part by grants from the NSF. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Past quarters: Fall 2013, Winter 2014, Spring 2014, Fall 2014, Fall 2017, Winter 2018, Spring 2018, Fall 2018, Winter 2019, Fall 2019, Fall 2019.
The design of this webpage is copied shamelessly from the MIT Number Theory seminar site. Contact Iacopo Brivio at ibrivio@ucsd.edu about problems with the website or posters.