Van Geemen-Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb{P}^2$

Abstract

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of P 2 branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen-Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen-Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.

Women in Numbers Europe

“…Indeed the torsion part of the Mordell-Weil group, which is already present in the more general fibrations analyzed in Comparin and Garbagnati (2014), are the same and the difference between the fibration 22 and the fibration 22 (b) is in the free part of the Mordell Weil group, so the difference between these two fibrations involves exactly the classes that correspond to our specialization.…”

“…The elliptic fibrations on the generic member Y of this family have already been classified (cf. Comparin and Garbagnati 2014), and indeed the elliptic fibrations in Table 1 specialize the ones in Comparin and Garbagnati (2014, Table 4.5 and Section 8.1 case r D 19), either because the rank of the Mordell-Weil group increases by 1 or because two singular fibers glue together producing a different type of reducible fiber.…”

“…The method proposed in Oguiso (1989) is very geometric: it is strictly related to the presence of a certain automorphism (a non-symplectic involution) on the K3 surface. Since one has to require that the K3 surface admits this special automorphism, the method suggested in Oguiso (1989) can be generalized only to certain special K3 surfaces (see Kloosterman 2006;Comparin and Garbagnati 2014). Seven years after the paper (Oguiso 1989), a different method was proposed by Nishiyama in Nishiyama (1996).…”

“…They satisfy Tr.E 9 / ' Tr.E 21 / and MW.E 9 / ' MW.E 21 /, but E 9 is not J 2 -equivalent to E 21 since, as above, the infinite order sections of these two fibrations have different intersection properties with the singular fibers. Indeed these two fibrations correspond to different fibrations (Comparin and Garbagnati 2014, Case 10a) and case 10b), Section 8.1, Table r D 19) on the more general family of K3 surfaces considered in Comparin and Garbagnati (2014). world junior and senior women, bringing their skill, experience, and knowledge from geometry and number theory.…”

Elliptic Fibrations on Covers of the Elliptic Modular Surface of Level 5

“…k for k = 1, 2, 3, 4, Q 2 and Q 4 and they span the lattice E 8 , so N 1 corresponds to the fibration 1 in Table (6.1); the curves orthogonal to N 2 are Ω (1) i,j , for i, j ∈ {0, 1, 2, 3, 4}, i < j, Ω (1) k for k = 1, 2, 3, 4 and Q 2 and they span the lattice D 10 , so N 2 corresponds to the fibration 2 in Table (6.1).…”