This is the homepage of the University of Illinois Urbana-Champaign Algebraic Geometry Seminar, organized by Deniz Genlik and Sheldon Katz. We meet Tuesdays at 3-3:50pm.
Other algebraic geometry activities at Illinois: AG lunch (Tu 12:30-1:30pm), AG preprint seminar (Tu 2-2:50pm). For more information, join our mailing list, either by visiting here or emailing Chris Dodd, Felix Janda, or Sheldon Katz.
Fall 2025 schedule:
September 2: Lena Ji (University of Illinois Urbana-Champaign)
Title: Finite order birational automorphisms of Fano hypersurfaces
Abstract: The birational automorphism group of an algebraic variety is an interesting birational invariant. For general type varieties this group is always finite, but for Fano varieties the situation is more complicated; for example, for P^n, the birational automorphism group is the mysterious Cremona group. In this talk, we study Fano hypersurfaces. We prove that there exist Fano hypersurfaces of arbitrarily high Fano index (in sufficiently high dimension) that admit no finite order birational automorphisms. A key input is the study of a specialization homomorphism for the birational automorphism group, which was defined by Matsusaka and Mumford. This work is joint with Nathan Chen and David Stapleton.
September 9: Ravi Fernando (University of Illinois Urbana-Champaign)
Title: Dimensional vanishing of the saturated de Rham-Witt complex
Abstract: The saturated de Rham-Witt complex, introduced by Bhatt-Lurie-Mathew, is a variant of the classical de Rham-Witt complex which is expected to behave better for singular schemes. We provide partial justification for this expectation by showing that the saturated de Rham-Witt complex satisfies dimensional vanishing even in the presence of singularities—like étale cohomology, but unlike any of de Rham, classical de Rham-Witt, and crystalline cohomology.
September 16: Christopher Dodd (University of Illinois Urbana-Champaign)
Title: Differential Operators, Gauges, and Mazur’s Theorem for Hodge Modules
Abstract: In this talk, I will explain how Mazur’s theorem about Frobenius and the Hodge filtration (from the early 1970’s) can be put into the context of filtered D-module theory. Along the way, I’ll explain the notion of a gauge, discuss Berthelot’s arithmetic differential operators, and explain how the Hodge filtration on a mixed Hodge module of geometric origin fits into the picture. I’ll give at least the basic background on all of the objects that appear in the title of the talk.
September 23: Ian Cavey (University of Illinois Urbana-Champaign)
Title: Newton-Okounkov body computations for M_{0,n}-bar
Abstract: Newton-Okounkov bodies (N.O. bodies) are convex sets that encode asymptotic information about sections of line bundles on a projective variety. In this talk, I will discuss the problem of computing N.O. bodies for line bundles on M_{0,n}-bar, the moduli space of stable pointed rational curves. I will describe an algebraic approach for computing N.O. bodies of arbitrary line bundles on M_{0,n}-bar, as well as a more geometric approach that in general produces only subset of the N.O. bodies. In some cases of interest, we conjecture that the geometric approach produces the entire N.O. body. This talk is based on upcoming joint work with Deniz Genlik.
September 30: Joshua Enwright (University of California, Los Angeles)
Title: Varieties of Small Complexity
Abstract: The complexity is an invariant of log pairs that was shown by Brown-McKernan-Svaldi-Zong to characterize toric varieties. More precisely, they showed that toric Calabi-Yau pairs minimize the complexity among all Calabi-Yau pairs. I will discuss two works that study this invariant further. The first, joint with Fernando Figueroa (Northwestern), identifies all minimizers of the complexity and studies their birational geometry. The second, joint with Jennifer Li (Princeton) and José Yáñez (UCLA) studies other geometric consequences of small complexity and provides a criterion in terms of the complexity for a variety to be cluster type.
October 7: David Zureick-Brown (Amherst College)
Title: The Canonical Ring of a Stacky Curve
Abstract: We give a generalization to stacks of the classical theorem of Petri - i.e., we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give explicit presentations of various rings of modular forms. This is joint work with John Voight.
Octover 14: Eric Zaslow (Northwestern University)
Title: New Counts of Nodal Curves
Abstract: I will describe work in progress falling somewhere between the count of nodal curves on K3 and Kontsevich’s famous count of rational plane curves of degree d meeting 3d-1 points. Specifically, I will try to count nodal curves in a toric Fano surface with fixed intersection with the toric boundary. In the case of P^2, this means genus-zero degree-d curves fixing 3d points along the three coordinate axes. While I can’t give a general formula (yet?), I can give a definition and present a couple of interesting (to me) ways that use a kind of mirror duality to approach the computation. Results agree (so far) for low degrees in many examples. This work is joint with Mingyuan Hu and Tom Graber.
October 21: Hsian-Hua Tseng (The Ohio State University) [Hybrid talk, see the note below.]
Title: On Hilb/Sym Correspondence
Abstract: For a smooth surface S, the Hilbert scheme of n points on S, Hilb(S,n), is smooth of dimension 2n. The Hilbert-Chow morphism Hilb(S,n)\to S^n/S_n is a crepant resolution of the (singular) n-fold symmetric product variety S^n/S_n associated to S. The so-called crepant resolution conjecture in Gromov-Witten theory predicts in this case explicit equalities between generating functions of Gromov-Witten invariants of Hilb(S,n) and generating functions of Gromov-Witten invariants of the symmetric product stack Sym(S,n). In this talk, we discuss the formulation of this conjecture and proofs in known cases.
Note: This is a hybrid talk. Audience will be physically present in the seminar room and the speaker will join our seminar room remotely.
October 28
November 4: Amy Q. Li (University of Texas Austin)
Title: TBA
Abstract: TBA
November 11: Bruce Reznick (University of Illinois Urbana-Champaign)
Title: Higher Order Arithmetic-Geometric Inequalities
Abstract: TBA If $\{\alpha_k\} \subset \mathbb R^d$ consists of a simplex and a single interior point, and if you impose the condition that $p(x) = \sum c_kx^{\alpha_k}$ vanishes to the second order at $\underbar{1} = (1,\dots,1)$ and is definite, then the resulting polynomial is, up to a multiple, a version of the arithmetic- geometric inequality for the monomials $\{x^{\alpha_k}\}$. There is a sense in which this is an iff condition. In this talk, we explore geometric conditions on larger point-sets $\{\alpha_k\}$ so that imposing a higher even-order vanishing at $\underbar{1}$ leads to an inequality, and present a few preliminary results and a lot of pictures. A riddle, wrapped in a mystery, inside an enigma.
November 18: Miguel Moreira (Massachusetts Institute of Technology)
Title: TBA
Abstract: TBA
December 2
December 9: Carl Lian (Washington University in St. Louis)
Title: TBA
Abstract: TBA