On a map between two K3 surfaces associated to a net of quadrics

Abstract

We study the geometry of the birational map between an intersection of a net of quadrics in $$ Q_1 \cap Q_2 \cap Q_3 \subset \user2{\mathbb{P}}_5 $$ that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S 6 is discriminant of a net of quadrics in $$ \user2{\mathbb{P}}_5 $$ such that S 6 is isomorphic to the intersection of this net iff a certain configuration of rational curves on S6 is weakly even.