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On certain curves of genus three in characteristic two. (English) Zbl 1143.14024
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.
14H20 Singularities of curves, local rings
14H45 Special algebraic curves and curves of low genus
14H51 Special divisors on curves (gonality, Brill-Noether theory)
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