 
Summary: CURVES OF GIVEN pRANK WITH TRIVIAL
AUTOMORPHISM GROUP
JEFFREY D. ACHTER, DARREN GLASS, AND RACHEL PRIES
Abstract. Let k be an algebraically closed field of characteristic p > 0.
Suppose g 3 and 0 f g. We prove there is a smooth projective
kcurve of genus g and prank f with no nontrivial automorphisms.
In addition, we prove there is a smooth projective hyperelliptic kcurve
of genus g and prank f whose only nontrivial automorphism is the
hyperelliptic involution. The proof involves computations about the
dimension of the moduli space of (hyperelliptic) kcurves of genus g and
prank f with extra automorphisms.
1. Introduction
Let k be an algebraically closed field of characteristic p > 0. If g 3, there
exist a kcurve C of genus g with Aut(C) = {1} and a hyperelliptic kcurve
D of genus g with Aut(D) Z/2 (see, e.g., [16] and [8], respectively). In this
paper, we extend these results to curves with given genus and prank.
If C is a smooth projective kcurve of genus g with Jacobian Jac(C), the
prank of C is the integer fC such that the cardinality of Jac(C)[p](k) is pfC
.
It is known that 0 fC g. We prove the following:
