On certain curves of genus three in characteristic two (Q2425547)

This paper considers curves of genus \(3\) over an algebraically closed field of characteristic \(2\), under the assumption that there exists a canonical theta characteristic represented by a divisor of the form \(2P_0\). These curves are non-hyperelliptic and they have a \(4\)-dimensional moduli space. The attention is focused on the \(2\)-dimensional subfamily formed by the curves that admit two different Weierstrass points \(Q_1,Q_2\) such that the divisors \(P_0+3Q_1\), \(P_0+3Q_2\) are canonical. Every such curve is isomorphic to a plane quartic with affine equation \[ C_{a,b,c}\colon x+y+ax^3y+bx^2y^2+cxy^3=0,\quad abc\neq0,\quad a+b+c\neq0. \] The paper contains an exhaustive analysis of the properties of the Weierstrass points of these curves.