arXiv subjects
Publications and source records attributed to Olof Bergvall.
We determine the number of Del Pezzo surfaces of degree 2 over finite fields of odd characteristic with specified action of the Frobenius endomorphism, i.e. we solve the "quantitative inverse Galois problem". As applications we determine the number of Del Pezzo surfaces of degree 2 with a given number of points and recover results of Banwait-Fit\'e-Loughran and Loughran-Trepalin.
We compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree three and four as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.
In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.
We consider moduli spaces of plane quartics marked with various structures such as Cayley octads, Aronhold heptads, Steiner complexes and Göpel subsets and determine their cohomology. This answers a series of questions of Jesse Wolfson. We also explore some arithmetic applications over finite fields.
We compute the cohomology of the complement of toric arrangements associated to root systems as representations of the corresponding Weyl groups. Specifically, we develop an algorithm for computing the cohomology of the complement of toric arrangements associated to general root systems and we carry out this computation for the exceptional root systems $G_2$, $F_4$, $E_6$ and $E_7$. We also compute the total cohomology of the complement of the toric arrangement associated to $A_n$ as a representation of the Weyl group and give a formula for its Poincaré polynomial.
We bound the dimensions of the graded pieces of the tautological ring of the universal curve from below for genus up to 27 and from above for genus up to 9. As a consequence we obtain the precise structure of the tautological ring of the universal curve for genus up to 9. In particular, we see that it is Gorenstein for these genera.
We determine the cohomology groups of the space of seven points in general linear position as representations of the symmetric group on seven elements by making equivariant point counts over finite fields. We also comment on the case of eight points.
We compute the total cohomology of the complement of the toric arrangement associated to the root system $A_n$ as a representation of the corresponding Weyl group via fixed point theory of a "twisted" action of the group. We also provide several proofs of an explicit formula for the Poincaré polynomial of the complement of the toric arrangement associated with $A_n$.
We compute the weighted Euler characteristic, equivariant with respect to the action of the symplectic group of degree six over the field of two elements, of the moduli space of principally polarized abelian threefolds together with a level two structure.
We compute cohomology of the moduli space of genus three curves with level two structure and some related spaces. In particular, we determine the cohomology groups of the moduli space of plane quartics with level two structure as representations of the symplectic group on a six dimensional vector space over the field of two elements. We also make the analogous computations for some related spaces such as moduli spaces of genus three curves with a marked points and strata of the moduli space of Abelian differentials of genus three.
We make cohomological computations related to the moduli space of genus three curves with symplectic level two structure by means of counting points over finite fields. In particular, we determine the cohomology groups of the quartic locus as representations of the symmetric group on seven elements.