18th Sinchon Workshop on
Algebraic Geometry
연세대학교, 과학관 225
2026년 6월 17일

Talk information

• 배영한: Cycles on Mumford's partial compactification of the universal abelian scheme

Abstract: Let A_g be the moduli space of principally polarized abelian varieties of dimension g and let X_g be the universal family over A_g. By Beauville and Deninger-Murre, the Chow ring of X_g admits a multiplicative splitting by weights. By van der Geer, the tautological ring of X_g has a complete description. What can one say about cycles on (partial) toroidal compactification of X_g and A_g? In this talk, we answer this question for Mumford's partial compatification of X_g and A_g. This is a joint work in progress with J. Feusi, A. Iribar López, and S. Molcho. 

• 이동협: On Rank 3 Quadratic Generators of Veronese Embeddings

Abstract: Many projective varieties are cut out, ideal-theoretically, by quadratic equations of rank at most four. For example, if the defining ideal of a projective variety is generated by the (2 \times 2) minors of a matrix of linear forms, then it is generated by quadratic equations of rank at most four. Rational normal scrolls and Veronese varieties are classical examples of this type. It is therefore natural to ask which projective varieties have defining ideals generated by quadratic equations of rank at most three. In this talk, we will introduce a construction of rank-three quadrics arising from decompositions of the form (L=A^{\otimes 2}\otimes B) of a very ample line bundle (L) on a projective scheme, and explain how this construction can be used to prove rank-three generation of homogeneous ideals. We will then study the geometry of the locus of rank-three quadratic equations. Finally, we discuss how the results depend on the characteristic of the base field.

• 최도영: Hilbert polynomials of the first and second secant varieties

Abstract: In this paper, we give a description of the cohomology groups of the symmetric powers of the tautological bundle associated with a sufficiently positive line bundle on the Hilbert scheme of 2 or 3 points on a smooth projective complex variety. The dimensions of these cohomology groups are expressed in terms of the Hilbert polynomials of the first and second secant varieties of the embedding given by the sufficiently positive line bundle. We also compute these Hilbert polynomials completely.