"A complex polarized K3 surface (X,H) is reflexive if (1) H2 = 2; (2) there exists a divisor L with L·H = 0, L2 = −12; (3) D·H > 2 for every nodal curve D in X. The authors prove the following interesting result: There is a coarse moduli space K parametrizing reflexive K3 surfaces, which is an irreducible quasi-projective scheme of dimension 18. K3 surfaces satisfying condition (1) (with H ample) are so-called degree 2 K3 surfaces since they are 2 : 1 covers of projective plane P^2 branched along a non-singular plane sextic curve. The moduli space K^a_M of ample M-polarized K3 surfaces (where M is a lattice of rank 1 generated by a vector e with e2 = 2) was first constructed by J. Shah [Ann. of Math. (2) 112 (1980), no. 3, 485–510; MR0595204 (82j:14030)] using GIT. The authors make use of the space K^a_M to show that there is a coarse moduli space K parametrizing reflexive K3 surfaces. The non-emptiness of K is proved by showing that a generic Kummer surface X is reflexive by exhibiting a polarization H and a divisor L on X satisfying conditions (1), (2) and (3). A generic Kummer surface is a Kummer surface associated to the Jacobian variety T = J(C) of a smooth genus 2 curve whose Néron-Severi group NS(T) is generated by the \Theta-divisor." Bangere P. Purnaprajna, MR1477439 (99e:14038)

Moduli of reflexive K3 surfaces

BARTOCCI, CLAUDIO;
1997-01-01

Abstract

"A complex polarized K3 surface (X,H) is reflexive if (1) H2 = 2; (2) there exists a divisor L with L·H = 0, L2 = −12; (3) D·H > 2 for every nodal curve D in X. The authors prove the following interesting result: There is a coarse moduli space K parametrizing reflexive K3 surfaces, which is an irreducible quasi-projective scheme of dimension 18. K3 surfaces satisfying condition (1) (with H ample) are so-called degree 2 K3 surfaces since they are 2 : 1 covers of projective plane P^2 branched along a non-singular plane sextic curve. The moduli space K^a_M of ample M-polarized K3 surfaces (where M is a lattice of rank 1 generated by a vector e with e2 = 2) was first constructed by J. Shah [Ann. of Math. (2) 112 (1980), no. 3, 485–510; MR0595204 (82j:14030)] using GIT. The authors make use of the space K^a_M to show that there is a coarse moduli space K parametrizing reflexive K3 surfaces. The non-emptiness of K is proved by showing that a generic Kummer surface X is reflexive by exhibiting a polarization H and a divisor L on X satisfying conditions (1), (2) and (3). A generic Kummer surface is a Kummer surface associated to the Jacobian variety T = J(C) of a smooth genus 2 curve whose Néron-Severi group NS(T) is generated by the \Theta-divisor." Bangere P. Purnaprajna, MR1477439 (99e:14038)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11567/372118
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