Del Pezzo surfaces of degree 4 and their relation to Kummer surfaces

Abstract

In this note, which has little pretence to originality, we clarify the relation between the geometry of del Pezzo surfaces of degree 4 and their realization as the zero set of two quadratic forms in five variables. We also review the classical description of the desingularized Kummer surface K constructed from the Jacobian J of a curve C of genus 2 as the zero set of three quadratic forms in six variables (Plucker, Kummer, Klein [7], [6], see [5] or [3] for a modern treatment). If C has a rational Weierstrass point, a partial diagonalization of this system gives rise to a natural projection onto a hyperplane, defining a finite morphism π : K → X of degree 2 onto a del Pezzo surface X of degree 4 (see [9, §6]). We show that X is the blow-up of Pk in the images of the five other Weierstrass points of C under the embedding of Pk as a conic in Pk. The morphism π sends the 16 lines on K to the 16 lines on X, and is equivariant with respect to the action of the subgroup of 2-division points J [2] ⊂ J . Thus π gives rise to a morphism from the twisted Kummer surface to the twisted del Pezzo surface. In our presentation it is obvious that all del Pezzo surfaces of degree 4 can be obtained in this way, an observation made by Victor Flynn in [4]. The fact that any 2-covering of J maps to a del Pezzo surface of degree 4 was first observed in [2], and used in [2], [1] and [9] to construct and visualize elements of order 2 in the Tate–Shafarevich group of J over Q using the theory of the Brauer– Manin obstruction on del Pezzo surfaces of degree 4. It was the author’s desire to understand the geometry behind these calculations that prompted him to write this note. I would like to thank Igor Dolgachev for useful discussions.