Homogeneous spaces and degree 4 del Pezzo surfaces

Abstract

Abstract. It is known that, given a genus 2 curve C : y 2 = f (x), where f (x) is quintic and defined over a field K, of characteristic different from 2, and given a homogeneous space H δ for complete 2-descent on the Jacobian of C, there is a V δ (which we shall describe), which is a degree 4 del Pezzo surface defined over K, such that H δ (K) = ∅ =⇒ V δ (K) = ∅. We shall prove that every degree 4 del Pezzo surface V , defined over K, arises in this way; furthermore, we shall show explicitly how, given V , to find C and δ such that V = V δ , up to a linear change in variable defined over K. We shall also apply this relationship to Hürlimann's example of a degree 4 del Pezzo surface violating the Hasse principle, and derive an explicit parametrised infinite family of genus 2 curves, defined over Q, whose Jacobians have nontrivial members of the ShafarevichTate group. This example will differ from previous examples in the literature by having only two Q-rational Weierstrass points.

Quadratic points on intersections of two quadrics

“…(3): The first statement is a theorem of Flynn [Fly09] which was expanded upon by Skorobogatov [Sko10]. The second statement follows from the first together with the Faddeev exact sequence for Br(k(P 1 )) (see [GS06,Thm 6.4.5] or (5.4)).…”

Anticanonical codes from del Pezzo surfaces with Picard rank one

“…Instead of computing a birational and F q -rational morphism from P 2 to the considered degree 5 or 6 del Pezzo surface, we adopt a different strategy in degree 4. In fact, it turns out that, following Flynn [Fly09], one can directly compute the anti-canonical model of a degree 4 del Pezzo surface from the Frobenius action on the geometric Picard group, at least when the characteristic is odd. This model, which is defined over the base field, is embedded in P 4 as the intersection of two quadrics.…”

“…Unlike the degrees 5 or 6 cases, there are several isomorphism classes in every type of Frobenius action. Flynn develops a very powerful method which, given a type of Frobenius action in the previous sense, computes the anti-canonical model of a del Pezzo surface having this type of action [Fly09,Sko10]. It works as follows, starting from a type d 1 [ǫ 1 ] • • • d r [ǫ r ].…”

Rational points on curves

“…In some cases, it is possible to show that there are non-trivial such elements, thereby improving the upper bound on the rank. Two techniques that have been suggested and also used are visualization [6] and the Brauer-Manin obstruction on certain related varieties [1,25,32].…”