Unirationality of del Pezzo surfaces of degree two over finite fields

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and V\'arilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.Comment: This is a short version (5 pages) of arXiv:1408.0269; the longer version contains more details and more general result

On the arithmetic of one del Pezzo surface over the field with three elements

We discuss the problem of existence of rational curves on a certain del Pezzo surface from a computational point of view and suggest a computer algorithm implementing search. In particular, our computations reveal that the surface contains 920 rational curves with parametrizations of degree 8 and does not contain rational curves for a smaller degree

Weak weak approximation and the Hilbert property for degree-two del Pezzo surfaces

We prove that del Pezzo surfaces of degree $2$ over a field $k$ satisfy weak weak approximation if $k$ is a number field and the Hilbert property if $k$ is Hilbertian of characteristic zero, provided that they contain a $k$-rational point lying neither on any $4$ of the $56$ exceptional curves nor on the ramification divisor of the anticanonical morphism. This builds upon results of Manin, Salgado--Testa--V\'arilly-Alvarado, and Festi--van Luijk on the unirationality of such surfaces, and upon work of the first two authors verifying weak weak approximation under the assumption of a conic fibration.Comment: 22 pages, minor edits, comments welcom

The unirationality of the moduli spaces of 2-elementary K3 surfaces (with an Appendix by Ken-Ichi Yoshikawa)

We prove that the moduli spaces of K3 surfaces with non-symplectic involutions are unirational. As a by-product we describe configuration spaces of 4<d<9 points in the projective plane as arithmetic quotients of type IV.Comment: 32 pages, simplified exposition, to appear in Proc. London Math. So

On the Unirationality of del Pezzo surfaces of degree two

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.Comment: 20 pages, Magma code included at the end of the source fil