Cecília Salgado

On the unirationality of del Pezzo surfaces of degree 2

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.

Classifications of Elliptic Fibrations of a Singular K3 Surface

We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface X whose transcendental lattice is isometric to \(\langle 6\rangle \oplus \langle 2\rangle\).

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

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, R5,5. Such surfaces have a natural elliptic fibration induced by the fibration on R5,5. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on R5,5. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type I5 and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.