Cohomology of moduli spaces of Del Pezzo surfaces

Arithmetic statistics on cubic surfaces

“…Even though Theorem 2.1 is stated for singular cohomology of the complex points, the same results hold (after tensoring with ℓ ) for H * ét (Y ( q ); ℓ ) for every q. Further, it is an important part of the results of Bergvall-Gounelas in [BG19] that H * (Y / PGL(4)) is minimally pure, i.e. that Frob q acts on H i ét by the scalar q −i .…”

“…that Frob q acts on H i ét by the scalar q −i . In fact, the argument in [BG19] in some sense reverses the steps here, to go from point count to etalé cohomology to singular cohomology.…”

“…This permutation of the 27 lines governs much of the arithmetic of the surface S: evidently the pattern of lines defined over q and, less obviously, the number of q points on S (or UConf n S etc). Work of Bergvall and Gounelas [BG19] allows us to compute the number of cubic surfaces over q where Frob q induces a given permutation, or rather a permutation in a given conjugacy class of W (E 6 ). The results in this paper can be thought of as a combinatorial (Theorem 1.1) or representation-theoretic (Theorem 2.3) reinterpretation of their computation.…”

Arithmetic and topology of classical structures associated to plane quartics

“…In [29], Jesse Wolfson proposes the problem of computing the cohomology of these spaces. The purpose of this note is to explain how these results can be derived from our earlier works [6,7,9,10] (see also [5,11,8,12]). From a more modern perspective, these classical structures are naturally understood in terms of level structures and subgroups of the symplectic group Sp(6, F 2 ).…”

Problems in Arithmetic Topology

“…More generally, as suggested in correspondence with Ravi Vakil, one can consider lines on del Pezzo surfaces of degree 1 ≤ d ≤ 9, where d = 3 is the case of cubic surfaces. For more details see, for example, [7] and [75].…”