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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.
##### MSC:
 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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