Arithmetic on Singular Del Pezzo Surfaces

Equivariant Compactifications of Two-Dimensional Algebraic Groups

“…For n ∈ N, a (−n)-curve (or simply a negative curve) on a non-singular projective surface is a rational curve with self-intersection number −n. On generalised del Pezzo surfaces, only (−1)-or (−2)-curves may occur (see [CT88,page 29]). Moreover, a generalised del Pezzo surface is ordinary if and only if it contains no (−2)-curves.…”

Del Pezzo surfaces of degree four

“…1) X has a Appoint iff a) X has a k-point for any place v of ^, and map is an isomorphism having a smooth Del Pezzo surface of degree 4 of rank 2 for its image, or its image is a singular intersection of two quadrics X' having two conjugate singularities and such that the line joining the singularities does not lie on X' (following [14] such a surface is called an Iskovskih surface). Let V be a smooth model of X'.…”

“…To conclude this section, let us return to tori and recall Voskresenskii's theorem ( [43], Ch. VI, § 7) which computes the cokernel of the homomorphism (14). Let G be a linear algebraic group over a global field A;, and let G(k) be the topological closure of the image of It is well known that a Del Pezzo surface of degree 4 is defined in IPf as a geometrically integral smooth intersection of two quadrics X== <%n Q^.…”

Manin’s conjecture for a singular sextic del Pezzo surface

“…Since S is singular normal with only rational double points, by [CT88,Prop. 0.1] an anticanonical divisor of S can be taken to be any divisor on S which pulls back to an anticanonical divisor on the minimal desingularisation S. The anticanonical embedding is a natural choice, for example in this embedding the lines are exactly the (−1)-curves and Manin's conjecture takes a simpler form.…”

“…Such surfaces have a well-known classification in terms of their singularity type and degree. See [Man86] and [CT88] for more information on smooth and singular del Pezzo surfaces respectively, and [Bro07] for a general overview of Manin's conjecture for del Pezzo surfaces.…”