You are cordially invited to the seminar organized by the Department of Mathematics.
Speaker: Sadık Terzi
“Hasse-Witt Invariants of Curves”
Abstract: Let X be a curve in positive characteristic p. The talk is based on the computation of the Hasse-Witt invariants(p-rank and a-number) of curves by using several methods. There is a vast literature on computing σ(X) by determining the action of Frobenius on the cohomology group H^1(X,OX) or equivalently the action of the Cartier operator on H^0(X,ΩX). In the former case one essentially determines the Hasse-Witt matrix ([3]) and in the latter the Cartier-Manin matrix ([6]) describing the action. For an extensive bibliography on Hasse-Witt and Cartier-Manin matrices we refer to [1]. In this talk, we are concerned with the computation of the Hasse-Witt invariants of curves in two different setups ([4]). We first consider a pair X,X′ of proper curves over the field Fp, where X′ is a singular curve which lies on a smooth surface S (with pg(S) = 0 = q(S))and X is the smooth model of X′. We determine the p-rank of X by using the exact sequence of group schemes relating the Jacobians JX and JX′ . Next, we work with complete intersection varieties in Pn for n ≥ 2 and compute explicitly the action of Frobenius on the top cohomology group. In case of curves and surfaces, this information suffices to determine if the variety is ordinary. The last part of the talk will provide a family of smooth curves, introduced by Hidalgo in [2], and a family of singular curves. In [5], we compute a-number and “p-rank” of smooth family, and we provide a relation for Hasse-Witt invariants between singular family and its smooth model by using method described in [4].
References
[1] J. D. Achter, and E. W. Howe Hasse–Witt and Cartier–Manin matrices: A warning and a request, arXiv:1710.10726v5 [math.NT], 2020.
[2] R. A. Hidalgo, Holomorphic differentials of generalized Fermat curves, J. Number Theory 217 (2020), 78–101.
[3] J. I. Manin, The Hasse–Witt matrix of an algebraic curve, Amer. Math. Soc. Transl. (2), 45:245–264, 1965. Translated by J.W. S. Cassels.
[4] S. Terzi, On the p-Rank of Curves, PAMS.
[5] S. Terzi, On the p-rank of singular curves and their smooth models, submitted.
[6] N. Yui, On the Jacobian varieties of hyperelliptic curves over fields of characteristic p > 2, J. Algebra, 52(2):378–410, 1978.
Date: 27 June 2024 Thursday
Time: 13:40
Place: SA-141