Related papers: A gem of the modular universe

We study the class of complex algebraic K3 surfaces admitting an embedding of H+E8+E8 inside the Neron-Severi lattice. These special K3 surfaces are classified by a pair of modular invariants, in the same manner that elliptic curves over…

We investigate the moduli spaces of one- and two-dimensional sheaves on projective K3 and abelian surfaces that are semistable with respect to a nongeneral ample divisor with regard to the symplectic resolvability. We can exclude the…

A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory quotients come with a natural ample line bundle, and hence often a natural projective embedding. This question…

We investigate the moduli of genus 10 curves that are endowed with a faithful action of the icosahedral group $\mathcal{A}_5$. We show among other things that this has the structure of a Deligne-Mumford stack whose underlying coarse moduli…

In this note we speculate about the structure of maximal product subvarieties in moduli stacks of Calabi-Yau manifolds. We discuss examples for quintic hypersurfaces in the four dimensional projective space.

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2\lambda_1+2\lambda_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic…

Shimura curves are moduli spaces of abelian surfaces with quaternion multiplication. Models of Shimura curves are very important in number theory. Klein's icosahedral invariants $\mathfrak{A},\mathfrak{B}$ and $\mathfrak{C}$ give the…

We give infinite series of groups Gamma and of compact complex surfaces of general type S with fundamental group Gamma such that 1) Any surface S' with the same Euler number as S, and fundamental group Gamma, is diffeomorphic to S. 2) The…

We explicitly construct the Kummer variety associated to the Jacobian of a hyperelliptic curve of genus 3 that is defined over a field of characteristic not equal to 2 and has a Weierstra{\ss} point defined over the same field. We also…

This paper studies curves on quartic K3 surfaces, or more generally K3 surfaces which are complete intersection in weighted projective spaces. A folklore conjecture concerning rational curves on K3 surfaces states that all K3 surfaces…

By work of Looijenga and others, one has a good understanding of the relationship between GIT and Baily-Borel compactifications for the moduli spaces of degree 2 K3 surfaces, cubic fourfolds, and a few other related examples. The…

Principally polarized abelian surfaces with prescribed real multiplication (RM) are parametrized by certain Hilbert modular surfaces. Thus rational genus 2 curves correspond to rational points on the Hilbert modular surfaces via their…

We study the higher Chow groups $CH^2(X,1)$ and $CH^3(X,2)$ of smooth, projective algebraic surfaces over a field of char 0. We develop a theoretical framework to study them by using so-called higher normal functions and higher…

We prove the modularity for an algebraic space curve $Y$ of genus $50$ in $\mathbb{P}^5$, which consists of $21$ quartic polynomials in six variables, by means of an explicit modular parametrization by theta constants of order $13$. This…

In this paper, we study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and we have natural…

We introduce a compact moduli scheme of marked noncommutative cubic surfaces as the GIT moduli scheme of relations of a quiver associated with a full strong exceptional collection on a cubic surface. It is a toric variety containing the…

We show that mod $2$ cohomological invariants of the moduli stack $\mathscr{M}_{3,n}$ of smooth pointed curves of genus three contain a free module with generators in degree $0$, $2$, $3$, $4$ and $6$, formed by the invariants of the…

Four level quantum systems, known as quartits, and their relation to two- qubit systems are investigated group theoretically. Following the spirit of Klein's lectures on the icosahedron and their relation to Hopf sphere bra- tions,…

In the present paper, we will show that three apparently disjoint objects: Galois representations arising from twenty-seven lines on a cubic surface (number theory and arithmetic algebraic geometry), Picard modular forms (automorphic…

We settle a long-standing question about the hypermultiplet moduli spaces of the heterotic strings on ALE singularities. These heterotic backgrounds are specified by the singularity type, an instanton number, and a (nontrivial) flat…