Back to Algebraic Geometry Seminar: Spring 2019
Algebraic Geometry Seminar
Date: Monday, Feb 11, 2019
Time: 3:30PM - 4:30PM
Location: LCB 222
(note special day) in JWB 208
Title |
Configurations of six-lines, string dualities, and modular forms |
Abstract |
A smooth K3 surface obtained as the blow-up of the quotient of a four-torus by the involution automorphism at all 16 fixed points is called a Kummer surface. Algebraic Kummer surfaces obtained from abelian varieties provide a fascinating arena for string compactification and string dualities as they are not trivial spaces but are sufficiently simple to analyze most of their properties in detail. However, their Picard rank is always bigger or equal 17, and they only provide a geometric description for heterotic string vacua with up to one Wilson line. In this talk, I give an explicit description of the family of K3 surfaces of Picard rank sixteen associated with the double cover of the projective plane branched along the union of six lines, and the family of its Van Geemen-Sarti partners, i.e., K3 surfaces with special Nikulin involutions, such that quotienting by the involution and blowing up recovers the former. The family of Van Geemen-Sarti partners is a four-parameter family of K3 surfaces with a H + E 7 (-1) + E 7 (-1) lattice polarization. We describe explicit Weierstrass models on both families using even modular forms on the bounded symmetric domain of type IV. We also show that our construction provides a geometric interpretation, called geometric two-isogeny, for the F-theory/heterotic string duality in eight dimensions with two Wilson lines. If time allows, I will also describe special six line configurations (such as six lines tangent to a conic, three lines meeting in a point) that correspond to Kummer surfaces of Jacobians of genus-two curves with principal polarization and those associated to (1, 2)-polarized abelian surfaces, as well as their applications in string theory and number theory. |
Algebraic Geometry