6/12 (Thur)
Yongnam Lee
Title: KSBA moduli of Catanese–Todorov surfaces and Kunev surfaces
Abstract: In the 1970s, Kunev constructed certain minimal surfaces of general type satisfying p_g = K^2 = 1, which gave a counterexample to the local Torelli theorem. Then Todorov (and Catanese independently) gave a description of all minimal surfaces of general type with p_g = K^2 = 1: the canonical model of such a surface is a weighted complete intersection of bidegree (6, 6) in the weighted projective space P(1, 2, 2, 3, 3). In particular, these surfaces can be viewed as degree 4 covers of the projective plane. The case in which this cover is Galois, recovers Kunev’s example. In this talk, I will talk about KSBA moduli of these surfaces, focusing in particular on boundary divisors.This work is in progress with Luca Schaffler.
Hirokazu Nasu
Title: Deformations of space curves lying on a del Pezzo surface
Abstract: Almost 40 years ago, J. O. Kleppe proposed a conjecture concerning maximal families of curves lying on a smooth cubic surface in P^3, which led to a generalization of Mumford’s example (1962) of a generically non-reduced, irreducible component of the Hilbert scheme. A version of this conjecture, modified by Ph. Ellia, asserts that curves in certain classes are obstructed and stably degenerate; that is, every small global deformation of such a curve remains contained in a deformation of a cubic surface, even though some of its infinitesimal deformations do not. We propose an analogous conjecture for space curves lying on a smooth del Pezzo surface of degree greater than 3, and we examine the degree 4 case in detail. As a consequence, we construct a new series of generically non-reduced components of the Hilbert scheme of P^4.
Insong Choe
Title: A wobbly divisor of nilpotency index greater than 2
Abstract: A vector bundle over an algebraic curve is said to be wobbly if it admits a nonzero nilpotent K-valued endomorphism. Drinfeld's conjecture predicts that the wobbly bundles form a divisor in the moduli space. In this talk, I would like to explain a construction of a wobbly divisor of nilpotency index greater than 2. This is based on a joint work with George H. Hitching.
Young-Hoon Kiem
Title: Generalized sheaf counting
Abstract: It is well known that many problems in algebraic geometry are reduced to finding vector bundles or sheaves with desired properties. To enumerate them, we construct their moduli spaces and apply intersection theory. To get a moduli space with an intersection theory, we have to pick a stability condition and delete unstable objects. In good circumstances where there are no strictly semistable sheaves, integrating cohomology classes against the (virtual) fundamental class gives us enumerative invariants like Donaldson invariant, Seiberg-Witten invariant and Donaldson-Thomas invariant. However when there are strictly semistable sheaves, the moduli space is an Artin stack on which integration doesn't make sense under current technology. Generalized sheaf counting is about finding a way to modify the moduli space of semistable sheaves to get a Deligne-Mumford stack by which an enumerative invariant can be defined. In this talk, I will report recent progresses on generalized sheaf counting on curves, surfaces and 3-folds.
6/13 (Fri)
Jongil Park
Title: On the Euler characteristic
Abstract: The Euler characteristic is one of the most elementary and familiar invariants to undergraduate students, even middle/highschool students, as well as professional mathematicians. In this talk, I'd like to review how mathematicians have developed the Euler characteristic and utilized it in geometry and topology.
Kiryong Chung
Title: Quartic curves in the quintic del Pezzo threefold
Abstract: Let $X$ be the quintic del Pezzo threefold. By adjunction formula, the general intersection $X$ with linear subspace $H$ of codimension two is an elliptic quintic curve $E_5$. If we choose the linear subspace $H$ containing a line lying in $X$, then $E_5$ is the union of that line and rational quartic curve meeting at two points. In this talk, we prove that each rational quartic curve takes arise in this way even though the curve may not be reducible. This is a parallel study with that of arXiv:2412.17721 and is a joint work with Jaehyun Kim and Jeong-Seop Kim.
Jinhyung Park
Title: Singularities and syzygies of secant varieties of smooth projective varieties
Abstract: The k-th secant variety of a smooth projective variety embedded in projective space is the Zariski closure of the union of the planes spanned by k+1 distinct points. Suppose that the embedding is given by the complete linear system of a sufficiently positive line bundle. About 10 years ago, Ullery and Chou-Song proved that the first secant variety has normal Du Bois singularities. About 5 years ago, in joint work with Ein and Niu, we generalized this result to higher secant varieties of curves, and showed that the k-th secant varieties of curves satisfy N_{k+2,p}-property meaning that the minimal free resolution of the section ring is as simple as possible until the p-th step. In this talk, I report recent joint work with Choi, Lacini, and Sheridan. We undertake a systematic study of secant varieties in all dimensions based on geometry of Hilbert schemes of points. In particular, we determine exactly when secant varieties have extra singularities not lying in the previous secant varieties, and we extend the previous results on singularities and syzygies of secant varieties in the case when the Hilbert scheme of k+1 points is smooth.
Euisung Park
Title: On Quadratic Equations of Curves of Genus Two
Abstract: In his 1970 paper, Saint-Donat proved that if a smooth projective curve C of genus g is embedded into a projective space by a line bundle of degree at least 2g+2, then the homogeneous ideal of its image is generated by quadratic polynomials of rank at most 4.
In this talk, we focus on the case when the genus is 2. Specifically, (i) when the degree is 6, we will describe the relationship between the Kummer surface associated to C and the locus of quadratic equations of rank at most 4; and (ii) when the degree is at least 6, we will explain that the homogeneous ideal is generated by quadratic polynomials of rank 3. This is based on joint work with Changho Han.
6/14 (Sat)
JongHae Keum
Title: Fake quadric surfaces
Abstract: A smooth projective complex surface S is called a Q-homology quadric if it has the same Betti numbers as the smooth quadric surface. Let S be a Q-homology quadric. Then its cohomology lattice is of rank 2, (even or odd) unimodular. By the classification of surfaces, S is either rational or of general type. In the latter case, S is called a fake Q-homology quadric. There is an unsolved question raised by Hirzebruch: does there exist a surface of general type which is homeomorphic to the smooth quadric surface? I will report recent progress on these surfaces.
Kyusik Hong
Title: Hypersurfaces in $\mathbb{P}^4$ with positive defect
Abstract: Let $V$ be a hypersurface in $\mathbb{P}^4$ with ordinary double and triple points as the only singularities, and let $\mu_2$ and $\mu_3$ be the number of ordinary double points (triple points, respectively) of $V$. Then we prove that if $\mu_2 + 11\mu_3 \geq \frac{11d^4-50d^3+85d^2-70d+48}{24}$, then the defect of $V$ is positive. This is based on a joint work with Seung-Jo Jung.
Hristo Iliev
Title: Multiple Coverings of Curves on Cones: Tschirnhausen Modules and Components of the Hilbert Scheme of curves
Abstract: Let $\varphi: X \to Y$ be a finite, flat morphism of degree $m \geq 2$, where $X$ and $Y$ are smooth, irreducible, projective curves. Such a morphism induces a short exact sequence of vector bundles on $Y$:
\[0 \to \mathcal{O}_Y \xrightarrow{\varphi^{\sharp}} \varphi_{\ast} \mathcal{O}_X \to \mathcal{E}^{\vee} \to 0 \, ,\]
where $\mathcal{E}^{\vee}$ is known as the \emph{Tschirnhausen module} of the covering.
In this talk, we consider the case where the covering curve $X$ lies on a cone $F$ with base curve $Y$. In such situations, we show that the Tschirnhausen module $\mathcal{E}^{\vee}$ is completely decomposable, with summands (up to twist) of \emph{Veronese type}. As an application, we construct new families of curves on cones that yield both generically smooth and non-reduced components of the Hilbert scheme of curves.
Jun-Muk Hwang
Title: Deformations of the tangent bundle of a projective hypersurface
Abstract: Let X be a nonsingular hypersurface of dimension at least three in the complex projective space. It is well-known that the tangent bundle of X is a stable bundle. We describe an explicit birational model of the irreducible component of the moduli space of stable bundles on X containing the tangent bundle of X. This is a joint-work with Insong Choe and Kiryong Chung.