On normal K3 surfaces (original) (raw)
Mathematische Zeitschrift
We prove the existence of (20 − 2K 2)-dimensional families of simply-connected surfaces with ample canonical class, p g = 1, and 1 ≤ K 2 ≤ 9, and we study the relation with configurations of rational curves in K3 surfaces via Q-Gorenstein smoothings. Our surfaces with K 2 = 7 and K 2 = 9 are the first surfaces known in the literature, together with the existence of a 4dimensional family for K 2 = 8. Contents 28 4.9. More surfaces for each 2 ≤ K 2 ≤ 9 29 References 35
Singularities of normal quartic surfaces III (char=2, non-supersingular)
Cornell University - arXiv, 2022
We show, in this third part, that the maximal number of singular points of a normal quartic surface X ⊂ P 3 K defined over an algebraically closed field K of characteristic 2 is at most 12, if the minimal resolution of X is not a supersingular K3 surface. We also provide a family of explicit examples, valid in any characteristic.
Kummer quartic surfaces, strict self-duality, and more
2021
In this paper we first show that each Kummer quartic surface (a quartic surfaceX with 16 singular points) is, in canonical coordinates, equal to its dual surface, and that the Gauss map induces a fixpoint free involution γ on the minimal resolution S of X . Then we study the corresponding Enriques surfaces S/γ. We also describe in detail the remarkable properties of the most symmetric Kummer quartic, which we call the Cefalú quartic. We also investigate the Kummer quartic surfaces whose associated Abelian surface is isogenous to a product of elliptic curves with kernel (Z/2), and show the existence of polarized nodal K3 surfaces X of any degree d = 2k with the maximal number of nodes, such that X and its nodes are defined over R. We take then as parameter space for Kummer quartics an open set in P, parametrizing nondegenerate (166, 166)-configurations, and compare with other parameter spaces. We end with some remarks on normal cubic surfaces.
24 rational curves on K3 surfaces
2019
Given d in IN, we prove that all smooth K3 surfaces (over any field of characteristic p different from 2,3) of sufficiently high degree contain at most 24 rational curves of degree at most d. In the exceptional characteristics, the same bounds hold for non-unirational K3 surfaces, and we develop analogous results in the unirational case. We also show that the bounds are sharp and attained only by K3 surfaces with genus one fibrations.
2021
We prove the existence of (20 − 2K)-dimensional families of simply-connected surfaces with ample canonical class, pg = 1, and 1 ≤ K ≤ 9, and we study the relation with configurations of rational curves in K3 surfaces via Q-Gorenstein smoothings. Our surfaces with K = 7 and K = 9 are the first surfaces known in the literature, together with the existence of a 4dimensional family for K = 8.
Dynkin diagrams of rank 20 on supersingular K3 surfaces
Science China Mathematics, 2014
We classify normal supersingular K3 surfaces Y with total Milnor number 20 in characteristic p, where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y. 2. The Néron-Severi lattices of supersingular K3 surfaces A free Z-module Λ of finite rank with a non-degenerate symmetric bilinear form Λ × Λ → Z is called a lattice. Let Λ be a lattice. The dual lattice Λ ∨ of Λ is the Z-module Hom(Λ, Z). Then Λ is naturally embedded into Λ ∨ as a submodule of finite index. There exists a natural Q-valued symmetric bilinear form on Λ
Transcendental Lattices of Certain Singular K3 Surfaces
2008
We present a method of Zariski-van Kampen type for the calculation of the transcendental lattice of a complex projective surface. As an application, we calculate the transcendental lattices of complex singular K3 surfaces associated with an arithmetic Zariski pair of maximizing sextics of type A10 + A9 that are defined over Q( √ 5) and are conjugate to each other by the action of Gal(Q( √ 5)/Q).