Leiden University Student Repository

Evangelatos, Charalampos (2024)

On the Genus of Moduli Spaces of Rank 19 Lattice Polarized K3 Surfaces with a Level Structure

Master thesis | Mathematics (MSc)

Moduli spaces of lattice polarized K3 surfaces were first (defined and) studied in detail by Igor Dolgachev in his paper on Mirror Symmetry for Polarized K3 surfaces. We expand on the moduli space of lattice polarized K3 surfaces by adding level structure. The polarization we are interested in is given by a lattice M_n , which is primitively embedded into the K3 lattice and is of rank 19. In this case, the moduli space of (ample) M_n-polarized K3 surfaces with level l-structure is a modular curve. We compute the genus of this moduli space when l is an odd power of a prime. To this goal, we deduce that this moduli space can be expressed as a quotient of the extended upper half plane with a subgroup of PSL(2,R). By computing this subgroup, and in turn, its index and number of inequivalent cusps we will use the Riemann-Hurwitz formula to determine its genus. Additionally, we extend this subgroup by the Fricke involution which gives rise to a new modular curve for which we compute...Show moreModuli spaces of lattice polarized K3 surfaces were first (defined and) studied in detail by Igor Dolgachev in his paper on Mirror Symmetry for Polarized K3 surfaces. We expand on the moduli space of lattice polarized K3 surfaces by adding level structure. The polarization we are interested in is given by a lattice M_n , which is primitively embedded into the K3 lattice and is of rank 19. In this case, the moduli space of (ample) M_n-polarized K3 surfaces with level l-structure is a modular curve. We compute the genus of this moduli space when l is an odd power of a prime. To this goal, we deduce that this moduli space can be expressed as a quotient of the extended upper half plane with a subgroup of PSL(2,R). By computing this subgroup, and in turn, its index and number of inequivalent cusps we will use the Riemann-Hurwitz formula to determine its genus. Additionally, we extend this subgroup by the Fricke involution which gives rise to a new modular curve for which we compute bounds of its genus.Show less

Author

Evangelatos, Charalampos

Faculty

Faculty of Science

Specialisation

Algebra, Geometry and Number Theory

Supervisors

Brakkee, E.L

ECTS Credits

40

Language

en