arXiv subjects
Publications and source records attributed to Matthias Schuett.
We prove that there are no effective or anti-effective classes of square -1 or -2 arising from nodal curves on smooth algebraic surfaces by way of divisibility. This general fact has interesting applications to Enriques and K3 surfaces. The proof relies on specific properties of root lattices and their duals.
We give a complete classification of complex Q-homology projective planes with isolated rational double point singularities and numerically trivial canonical bundle. There are 31 types, and each has one-dimensional moduli. In fact, all moduli curves are rational and defined over Q, and we determine all families explicitly using extremal rational elliptic surfaces and Enriques involutions of base change type.
For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.
Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a smooth quartic.
Over a field k of characteristic 3, we prove that there are no smooth quartic surfaces S in IP^3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent over k-bar to the Fermat quartic. As a key ingredient, we derive a characteristic free upper bound for the number of lines met by a quadric on a smooth quartic surface.
We solve the Picard number problem for complex quintic surfaces by proving that every number between 1 and 45 occurs as Picard number of a quintic surface over the rationals. Our main technique consists in arithmetic deformations of Delsarte surfaces, but we also use K3 surfaces and wild automorphisms.
The supersingular K3 surface X in characteristic 2 with Artin invariant 1 admits several genus 1 fibrations (elliptic and quasi-elliptic). We use a bijection between fibrations and definite even lattices of rank 20 and discriminant 4 to classify the fibrations, and exhibit isomorphisms between the resulting models of X. We also study a configuration of (-2)-curves on X related to the incidence graph of points and lines of IP^2(IF_4).
This paper concerns K3 surfaces with automorphisms of order 11 in arbitrary characteristic. Specifically we study the wild case and prove that a general such surface in characteristic 11 has Picard number 2. We also construct K3 surfaces with an automorphism of order 11 in every characteristic, and supersingular K3 surfaces whenever possible.
Double planes branched in 6 lines give a famous example of K3 surfaces. Their moduli are well understood and related to abelian fourfolds of Weil type. We compare these two moduli interpretations and in particular divisors on the moduli spaces. On the K3 side, this is achieved with the help of elliptic fibrations. We also study the Kuga-Satake correspondence on these special divisors.
For every known Hecke eigenform of weight 3 with rational eigenvalues we exhibit a K3 surface over QQ associated to the form. This answers a question asked independently by Mazur and van Straten. The proof builds on a classification of CM forms by the second author.
We study the geometry of quartic surfaces in IP^3 that contain a line of the second kind over algebraically closed fields of characteristic different from 2,3. In particular, we correct Segre's claims made for the complex case in 1943.
In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.
We prove that the Consani-Scholten quintic, a Calabi-Yau threefold over QQ, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livne method to induced four-dimensional Galois representations over QQ. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.
We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarised abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in IP^4 which is the complete intersection of two S_5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.
This paper investigates a specific smooth quintic surface suggested by Barth for it contains the current record of 75 lines over the complex numbers. Our main incentive is to prove that the complex quintic has Picard number 41, and to compute the Neron-Severi group up to a 2-power index. We also compute Picard numbers for reductions to positive characteristic and verify the Tate conjecture.
We give a geometric construction of the three infinite series of K3 surfaces which are sandwiched by Kummer surfaces within a Shioda-Inose structure. Explicit examples are also provided.
We prove the modularity of Maschke's octic and two Calabi-Yau threefolds derived from it as double octic and quotient thereof by the Heisenberg group, as conjectured by Bini and van Geemen. The proofs rely on automorphisms of the varieties and isogenies of K3 surfaces.
This paper investigates the Picard numbers of quintic surfaces. We give the first example of a complex quintic surface in IP^3 with maximum Picard number 45. We also investigate its arithmetic and determine the zeta function. Similar techniques are applied to produce quintic surfaces with several other Picard numbers that have not been achieved before.