The p-rank stratification of the moduli space of double covers of a fixed elliptic curve

In this paper we investigate the p-rank stratification of the moduli space of curves of genus g that admit a double cover to a fixed elliptic curve E in characteristic p>2. We show that the closed p-rank strata of this moduli space are equidimensional of the expected dimension. We also show the existence of a smooth double cover of E of all the possible values of the p-rank on this moduli space.

Deepesh Singhal
On a conjecture of Ghorpade, Datta and Beelen for the number of points of varieties over finite fieldsConsider a finite field Fq and positive integers d,m,r with 1≤ r≤ \binom{m+d}{d}. Let S_d(m) be the Fq vector space of all homogeneous polynomials of degree d in X₀,…,X_m. Let e_r(d,m) be the maximum number of 𝓆-rational points in the vanishing set of W as W varies through all subspaces of S_d(m) of dimension r. Ghorpade, Datta and Beelen had conjectured an exact formula of e_r(d,m) when q≥ d+1. We prove that their conjectured formula is true when q is sufficiently large in terms of m,d,r. The problem of determining e_r(d,m) is equivalent to the problem of computing the r^th generalized hamming weights of projective the Reed Muller code PRM_q(d,m). It is also equivalent to the problem of determining the maximum number of points on sections of Veronese varieties by linear subvarieties of codimension r. In the most recent version of the work, we proved the exact formula in special case where m=2.