Rational double points on supersingular 𝐾3 surfaces

Abstract

Abstract. We investigate configurations of rational double points with the total Milnor number 21 on supersingular K3 surfaces. The complete list of possible configurations is given. As an application, we also give the complete list of extremal (quasi-)elliptic fibrations on supersingular K3 surfaces.

MODULI CURVES OF SUPERSINGULAR K3 SURFACES IN CHARACTERISTIC 2 WITH ARTIN INVARIANT 2

“…A marking of a polarized supersingular K3 surface (X G , L G ) of type ( ) is a bijective map γ : P → Z(dG). [λ,ωλ,ωλ +ω] P6 = C (12) [λ +ω,ωλ +ω,ωλ] P7 = T (02, 12,22) [1,ω,ω] P8 = C (11) [λ + 1, 1, λ] P9 = T (02, 11,20) [1,ω, ω] P10 = C (10) [1, ωλ + 1, 0] P11 = T (02, 10,21) [1,ω, 0] P12 = T (10,11,12) [ [0, 1, ω] P19 = T (00, 11,22) [0, 1, 1] P20 = T (00, 10,20) [0, In particular, Aut(X GB [α] , L GB [α] ) is isomorphic to the extended Heisenberg group of order 18.…”

“…In [10], we have shown that every supersingular K3 surface X in characteristic 2 has a polarization of type ( ), and that, if L is a polarization of type ( ) on X, then the morphism Φ |L| is purely inseparable. In [11], we have constructed a nine-dimensional moduli space M of polarized supersingular K3 surfaces of type ( ).…”

“…[λ, 0, λ +ω] P14 = T (01,12,20) [1, 0, ω] P15 = T (01, 11,21) [1, 0, 1] P16 = T (01, 10,22) [1, 0, 0] P17 = C(00)…”

Unirationality of certain supersingular K3 surfaces in characteristic 5

“…In [19], we have shown that a supersingular K3 surface in characteristic 2 is birational to a normal K3 surface with 21A 1 -singularities, and that such a normal K3 surface is a purely inseparable double cover of P 2 . In [20], we have proved that a supersingular K3 surface in characteristic 3 with Artin invariant ≤ 6 is birational to a normal K3 surface with 10A 2 -singularities, and it is also birational to a purely inseparable triple cover of P 1 × P 1 .…”

On Kummer type construction of supersingular K3 surfaces in characteristic 2

“…In [Shimada 2004a;2004b], we showed that every supersingular K 3 surface in characteristic 2 is birational to a purely inseparable double cover of ‫ސ‬ 2 with 21 ordinary nodes, and we studied the Néron-Severi lattice of such a surface. Using the results obtained in [Shimada 2004b], we determined in [Shimada 2006] the moduli curve of polarized supersingular K 3 surfaces with Artin invariant ≤ 2 and with 21 ordinary nodes.…”