Choi, Youngook (Yeungnam Univ.)

Title: Hilbert Scheme of smooth projective curves on ruled surfaces

Abstract : Let $\mathcal{I}_{d,g,r}$ be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this talk, we discuss on a regular component which is different from the distinguished component of $\mathcal{I}_{d,g,r}$ dominating the moduli space $\mathcal{M}_g$.

Lee, Kyoungseog (CGP-IBS)

Title : Cox rings of surfaces of general type

Abstract : Cox ring is one of the most important tools in algebraic geometry. In the first part of this talk, I will review basic definitions and facts about Cox rings. Then I will discuss Cox rings of minimal surfaces of general type with p_g = 0. This talk is based on joint works with JongHae Keum.

Lee, Sanghyeon (KIAS)

Title : Hyperplane property of the genus one Gromov-Witten invariants for complete intersections

Abstract : In this talk, I will introduce an algebraic proof of Zinger's theorem comparing standard and reduced genus one Gromov-Witten invariants for all compact symplectic manifolds in the case of complete intersections of dimension 2 and 3 in projective spaces. The proof contains local structures of the genus one stable map space, and methods of localized chern character for 2-periodic complexes.

This is a generalization of the result of H. -L. Chang and J. Li, which deal with quintic threefold case. This is a joint work with Jeongseok Oh (Arxiv : 1809.10995).

Park, Jinhyung (KIAS)

Title: Normality and projective normality of higher secant varieties of curves

Abstract: Consider a smooth projective curve embedded in a projective space by the complete linear system of a line bundle of sufficiently large degree. The k-th secant variety is the union of (k+1)-secant k-planes to the curve. In this talk, I prove that all higher secant varieties have normal singularities and are projectively normal. Actually, I show that the projective normality of m-th secant varieties for all m<k implies the normality of k-th secant variety and the normality of k-th secant variety implies the projective normality. This is joint work with Lawrence Ein and Wenbo Niu.

Park, Kyeongdong (CGP-IBS)

Title: Severi varieties and smooth projective symmetric varieties

Abstract : Hartshorne conjectured and Zak proved that any smooth nondegenerate projective variety of dimension n embedded in P^N with N < 3n/2 + 2 satisfies Sec(X) = P^N, where Sec(X) denotes the secant variety of X. Severi varieties are the extremal case whose secant varieties are not the whole projective space. Zak classified that there are only four Severi varieties and realized that all of them are homogeneous. They arise in numerous geometric contexts and admit a uniform interpretation as the varieties of the rank 1 matrices of complex Jordan algebras consisting of 3 by 3 Hermitian matrices over the four division algebras. Furthermore, they have an interesting property that a generic hyperplane section is still homogeneous. I will explain how generic hyperplane sections of Severi varieties relate to smooth projective symmetric varieties of Picard number 1 whose restricted root systems are of type A2. Note that these varieties can be summarized in the geometric Freudenthal-Tits magic square associated to the exceptional Lie algebras.