The Algebraic Geometry Seminar takes place on Wednesdays 3:30–4:30PM. Pre-talks (which graduate students are particularly encouraged to attend) take place Wednesdays 3:00–3:25PM.
The in-person talks will take place in STON 215. Some talks will be presented online via Zoom.
Please see below for the most up to date information. Here is the Zoom link for the online talks:
https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1
August 27: Takumi Murayama (Purdue University)
Location: STON 215
Title: Effective Fujita-type theorems for surfaces in arbitrary characteristic
Abstract: Fujita’s conjecture states: If L is an ample divisor on a smooth projective variety X of dimension n, then K+(n+1)L is globally generated and K+(n+2)L is very ample. An affirmative answer to Fujita’s conjecture would give an effective way to embed smooth projective varieties in projective space given only the datum of an ample divisor. While recent work of Gu, Zhang, and Zhang shows that Fujita’s conjecture is false over fields of positive characteristic, it remains an open question whether there exist uniform constants a and b only depending on n such that aK+bL is globally generated or very ample. Such constants are known to exist over fields of characteristic zero.
In this talk, I will present my recent progress on this problem for surfaces over fields of arbitrary characteristic. I will also present my approach to this problem in arbitrary dimensions that reduces to finding lower bounds for Seshadri constants of divisors of the form cK+dL.
September 3: Zhiyuan Jiang (Purdue University)
Location: STON 215
Title: Birational Geometry and the Abundance Conjecture for Kähler Varieties
Abstract: One of the most important open problems in birational geometry is the abundance conjecture: If X is a minimal projective manifold, then KX is semiample. The abundance conjecture is known in dimensions up to 3, and an affirmative answer to the conjecture would imply many major conjectures in algebraic geometry, like invariance of plurigenera (which is now a theorem due to Siu) and Iitaka's conjecture. It is expected that, more generally, abundance conjecture should also hold for Kähler varieties. As one application, abundance for Kähler varieties, together with the minimal model program, would imply invariance of plurigenera for Kähler varieties (which is still open). For Kähler threefolds, the abundance conjecture and the minimal model program hold by the work of Campana, Das, Hacon, Höring, Păun, Peternell, and Ou. In this talk, I will discuss my new approach to the abundance conjecture for Kähler varieties, which also works in higher dimensions under certain assumptions.
September 10: Donu Arapura (Purdue University)
Location: STON 215
Title: Euler characteristics of Kollár-hyperbolic varieties
Abstract: A Kollár-hyperbolic variety is a normal (complex) projective variety such that the restriction of the fundamental group to all subvarieties are highly nontrivial in a suitable sense. The notion is due to Kollár, but I will take the credit/blame for the name. Examples of such varieties include (a) finite branched covers of abelian varieties, (b) compact quotients of hermitian symmetric domains, and (c) Kodaira fibrations. I want to explain a conjecture of my mine about cohomology of mixed Hodge modules on Kollár-hyperbolic varieties. Currently I can prove it for varieties of type (a) and (c). But since I probably won’t have time to prove anything, I will probably just explain the conjecture and some of its consequences.
September 17: Fanjun Meng (University of California, San Diego)
Location: STON 215
Title: Wall crossing for moduli of stable pairs
Abstract: Hassett showed that there are natural reduction morphisms between moduli spaces of weighted pointed stable curves when we reduce weights. I will discuss some joint work with Ziquan Zhuang which constructs similar morphisms between moduli of stable pairs in higher dimension.
September 24: Hannah Larson (University of California, Berkeley)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Brill–Noether theory of special curves
Abstract: Brill–Noether theory studies the maps of curves C to projective spaces. The classical Brill–Noether theorem (established by work of Eisenbud, Fulton, Geiseker, Griffiths, Harris, Lazarsfeld) describes the geometry of this space of maps when C is a general curve. However, the theorem fails for special curves, notably curves that are already equipped with some unexpected map to a projective space. The first case of this is when C is a low-degree cover of the projective line. For general such covers, the Hurwitz–Brill–Noether theorem (joint with E. Larson and I. Vogt) provides a suitable analogue. I'll also present recent results (joint with S. Vemulapalli) regarding the next natural case: when C is equipped with an embedding in the projective plane.
October 1: No Seminar
October 8: Wanchun (Rosie) Shen (Harvard University)
Location: STON 215
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October 15: Siqing Zhang (Yale University)
Location: STON 215
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October 22: Riku Kurama (University of Michigan)
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October 29: Florin Ambro (Simion Stoilow Institute of Mathematics of the Romanian Academy)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
Title: Succesive minima of line bundles
Abstract: The Seshadri constant of a polarized variety (X,L) at a point x measures how positive is the polarization L at x. If x is very general, the Seshadri constant does not depend on x, and captures global information on X. Inspired by ideas from the Geometry of Numbers, we introduce in this talk successive Seshadri minima, such that the last one is the Seshadri constant at a point, and the first one is the width of the polarization at the point. Assuming the point is very general, we obtain two results: a) the product of the successive Seshadri minima is proportional to the volume of the polarization; b) if X is toric, the i-th successive Seshadri constant is proportional to the i-th successive minima of a suitable 0-symmetric convex body. Based on joint work with Atsushi Ito.
November 5: Casimir Kothari (University of Chicago)
Location: STON 215
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November 12: Andreas Höring (Université Côte d’Azur)
Location: Zoom (https://purdue-edu.zoom.us/j/99467898661?pwd=FOMrspNCAywfyh5VFOKDS80aWZLZuX.1)
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November 19: Botong Wang (University of Wisconsin-Madison)
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December 3: Shreya Sharma (University of South Carolina, Columbia)
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December 10: Lingyao Xie (University of California, San Diego)
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March 11: Marta Benozzo (Université Paris-Saclay)
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If you are interested in giving a talk, please contact one of the organizers: