Fall 2024
September 3: Lena Ji (University of Illinois Urbana-Champaign)
Title: Some arithmetic and birational properties of pencils of quadrics
Abstract: The linear spaces contained in the base locus of a pencil of quadrics encode a lot of interesting geometry. For example, for pencils of even-dimensional quadrics, there is a deep relationship between these linear spaces and hyperelliptic curves, dating back to Weil. This has found numerous applications, e.g. to rational points and to moduli theory. In this talk, we study rationality questions for the Fano schemes of these linear spaces, especially over non-closed fields. Our main focus is the case of second maximal linear subspaces, and we generalize results of Hassett–Tschinkel, Benoist–Wittenberg, and Hassett–Kollár–Tschinkel. This work is joint with Fumiaki Suzuki.
September 10: Deniz Genlik (University of Illinois Urbana-Champaign)
Title: Holomorphic Anomaly Equations and Crepant Resolution Correspondence for C^n/Z_n
Abstract: In this talk, we present results on the higher genus Gromov-Witten theory of C^n/Z_n by examining its cohomological field theory structure in detail. Holomorphic anomaly equations are recursive partial differential equations predicted by physicists for a Calabi-Yau threefold. We prove holomorphic anomaly equations for C^n/Z_n for any n >= 3. In other words, we demonstrate a phenomenon of holomorphic anomaly equations in arbitrary dimensions, extending beyond the scope considered by physicists. The proof relies on showing that the Gromov-Witten potential of C^n/Z_n lies in a specific polynomial ring. Furthermore, we prove a crepant resolution correspondence for arbitrary genera for C^n/Z_n by showing that its cohomological field theory aligns with that of KP^{n-1}, where KP^{n-1} is the total space of the canonical bundle of P^{n-1}. More precisely, we demonstrate that the Gromov-Witten potential of KP^{n-1} also resides in a similar polynomial ring, and we establish that it matches with the Gromov-Witten potential of C^n/Z_n under an isomorphism of these polynomial rings. This talk is based on joint work with Hsian-Hua Tseng, as detailed in arXiv:2301.08389 and arXiv:2308.00780.
September 17: Sheldon Katz (University of Illinois Urbana-Champaign)
Title: Enumerative invariants of noncommutative Calabi-Yau threefolds
Abstract: In this talk, I consider a class of nodal Calabi-Yau threefolds X which do not admit Kahler small resolutions. Nevertheless, topological string amplitudes can be associated to analytic small resolutions of X with topologically nontrivial B-fields. These amplitudes can be computed by B-model techniques, leading to integer-valued enumerative invariants. These X also admit noncommutative resolutions. A theory of Gopakumar-Vafa invariants of a larger class of noncommutative Calabi-Yau threefolds is introduced, which are conjectured to agree with the enumerative invariants computed by the B-model techniques. This talk is based on arXiv:2212.08655, arXiv:2307.00047, and work in progress.
September 24: Patrick Lei (Columbia University)
Title: Higher genus Gromov-Witten invariants of smooth Calabi-Yau threefolds in weighted P^4
Abstract: I will describe a proof of the Yamaguchi-Yau finite generation conjecture for Gromov-Witten potentials of smooth Calabi-Yau hypersurfaces in weighted projective spaces P(1,1,1,1,2), P(1,1,1,1,4), and P(1,1,1,2,5) based on recent work arXiv:2409.11660 and arXiv:2409.11659.
October 1: no seminar
October 8: Bruce Reznick (University of Illinois Urbana-Champaign)
Title: Denominators in Hilbert's 17th Problem
Abstract: A real form of degree d in n variables is “psd" if it only takes non negative values, and is “sos” if it can be written as a sum of squares of forms of degree d/2; sos implies psd. But Hilbert proved in 1888 that there are psd ternary sextic forms ((n,d) = (3,6)) which are not sos and in 1893 proved that every psd ternary form is a sum of squares of rational functions. (On taking the common denominator, if p is psd then there exists F so that F^2p is sos.) His 17th problem was to prove this for n > 3, which Artin did in the 1920s in a non-constructive way. The first explicit examples of psd-not-sos forms didn’t appear until the 1960s.
I’ll survey what is known about denominators. For example, if p is a psd form which is strictly definitive (p(u) > 0 for u in S^{n-1}), then for a computable large N, (\sum_j x_j^2)^N*p is sos. On the other hand, there exist psd forms p with the property that every odd power of p is not sos. The proofs will be elementary.
October 15: Younghan Bae (University of Michigan)
Title: Intersection theory on logarithmic Picard group
Abstract: Let C/B be a family of nodal curves of genus g with n sections. The relative Jacobian parametrizing multi-degree 0 line bundles is not proper over the base B. Classically, one can compactify the relative Jacobian using stable rank 1 torsion-free sheaves, which depends on a choice of stability condition. Recently, Molcho and Wise constructed the logarithmic Picard group, providing a canonical compactification. In this talk, we introduce the logarithmic tautological subring of the logarithmic Picard group inside the logarithmic Chow group. We prove that the proper pushforward to the base preserves the logarithmic tautological ring.
When the base is the moduli space of stable curves, we propose a conjectural closed formula for the pushforward of monomials of divisor classes. This conjecture is motivated by an explicit computation of Arinkin’s kernel over the compactified Jacobians. Over the open substack of compact type or integral curves, we prove the conjecture using the Fourier transformation. This is a joint work in progress with Samouil Molcho and with Aaron Pixton.
October 22: Alessio Cela (University of Cambridge)
Title: The Integral Chow Ring of Hyperelliptic Prym Pairs
Abstract: In this talk, I will introduce a parametrization of the moduli stacks of hyperelliptic Prym pairs and explain how to use equivariant intersection theory to compute their integral Chow rings.
October 29: Kirill Magidson (Northwestern University)
Title: A new approach to the De Rham-Witt complex
Abstract: I will present a new approach to the De Rham-Witt complex inspired by the work of Antieau and Nikolaus. The approach is based on the notion of differential graded Cartier modules which are an algebraic analogue of topological Cartier modules of Antieau-Nikolaus. The linear algebra of differential graded Cartier modules is only mildly more complicated than classical F-V complexes, and the new approach allows for a more conceptual and elegant revision of classical topics such as the crystalline comparison for the De Rham-Witt complex, and some new ones, such as the relation between De Rham-Witt connections and F-gauges on a smooth characteristic p scheme.
November 5: Jennifer Li (Princeton University)
Title: Rational surfaces with a non-arithmetic automorphism group
Abstract: In [Tot12], Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our examples Y are log Calabi-Yau surfaces, i.e., there is a reduced normal crossing divisor D in Y such that K_Y+D=0. This is joint work with Sebastián Torres.
November 12 (special NT/AG seminar): Ziquan Zhuang (Johns Hopkins University)
Special time and place: 11-11:50am, Gregory Hall room 205 (in the number theory seminar)
Title: Boundedness of singularities and discreteness of local volumes
Abstract: The local volume of a Kawamata log terminal (klt) singularity is an invariant that plays a central role in the local theory of K-stability. By the stable degeneration theorem, every klt singularity has a volume preserving degeneration to a K-semistable Fano cone singularity. I will talk about a joint work with Chenyang Xu on the boundedness of Fano cone singularities when the volume is bounded away from zero. This implies that local volumes only accumulate around zero in any given dimension.
November 12 (regular seminar): Fernando Figueroa (Northwestern University)
Title: Algebraic Tori in the Complement of Quartic Surfaces
Abstract: Log Calabi-Yau pairs can be thought of as generalizations of Calabi-Yau varieties. Previously Ducat showed that all coregularity 0 log Calabi-Yau pairs (P^3, S) are crepant birational to a toric model. A stronger property to ask for is for the complement of S to contain a dense algebraic torus. In that case, we say the pair (P^3, S) is of Cluster type.
In this talk we will show a full classification of coregularity zero, slc, reducible quartic surface for which their complements contain a dense algebraic torus. Along the way we will talk about relative cluster type pairs. We will finish by showing some partial results in the case of irreducible quartic surfaces.
This is based on Joint work with Eduardo Alves da Silva and Joaquín Moraga.
November 19: Stephen Pietromonaco (University of Michigan)
Title: Curve Counting for Abelian Surface Fibrations and Modular Forms
Abstract: A long-standing prediction of string theory and mirror symmetry is that certain formal generating series of curve-counting invariants are in fact expansions of quasi-modular objects. In this talk I will discuss on-going work with Aaron Pixton aiming to understand this modularity for Calabi-Yau threefolds fibered by Abelian surfaces (of Picard rank 2 or 3). I will focus on two very explicit examples: the banana manifold and Schoen nano-manifold. In both cases, we are interested in the Gromov-Witten (GW) potentials F_{g,k} where we assemble into a generating series the GW invariants of genus g for curve classes of degree k over the base. For the banana manifold, the GW potentials are formal series in 19 variables, which we conjecture to be Siegel-Jacobi forms for the E_{8} lattice, as introduced by Ziegler in the late 80s. As evidence, we prove an elliptic transformation law. For the Schoen nano-manifold, the GW potentials are formal series in only 2 variables, which we conjecture to be symmetric tensor products of quasi-modular forms with level. In degree k=1, this is consistent with work of Bryan-Oberdieck via degeneration to a CHL model.
December 3: Ben Castle (University of Illinois Urbana-Champaign)
Title: Reconstructing Abelian Varieties using Model Theory
Abstract: In 2012, Zilber used model theory to answer a question of Bogolomov, Korotaev, and Tshinkel. Essentially, he showed the following: let K be an algebraically closed field, let C be a smooth, irreducible, projective curve over K of genus at least 2, and let J be the Jacobian of C, viewed as an algebraic group containing C (via a fixed embedding). Then C is determined in the strongest sense possible by the data (J(K),+,C(K)) (as an abstract group with a distinguished subset). The proof of Zilber’s theorem relied on the `curve case’ of a model-theoretic conjecture, and this conjecture was recently proven in all dimensions. Thus, one can now consider higher-dimensional settings for Zilber’s theorem. This talk will present such a statement. Namely, suppose A is an abelian variety over an algebraically closed field, and V is a closed, irreducible subvariety; I will characterize when the pair (A,V) is determined in the strongest possible sense by the `abstract group with subset’ (A(K),+,V(K)).