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Weierstrass points of weight two on curves of genus three. (Thesis). (English) Zbl 0534.14010
Amsterdam: Universiteit van Amsterdam, Faculteit voor Wiskunde en Natuurwetenschappen. III, 183 p. (1983).
The aim of the paper is the description of the stratification given by Weierstrass points on the moduli space $$M_ 3$$ of the isomorphism classes of Riemann surfaces of genus 3. The weighted number of Weierstrass points on such a Riemann surface is 24 and the weight can be 1, 2 or 3. A hyperelliptic curve of genus 3 has exactly 8 Weierstrass points of weight 3 and a curve containing at least a point of weight 3 is necessarily hyperelliptic. A non hyperelliptic curve of genus 3 can be embedded in $${\mathbb{P}}^ 2$$ as curve of degree 4 and its Weierstrass points are exactly the points of inflexion. Thus the core of the question will be the study of the inflexion points of smooth plane curves of degree 4. A point of inflexion p on such a curve C with tangent line $$\ell$$ is called an ordinary flex resp. a hyperflex) if $$\ell$$ intersects C at p with multiplicity three (resp. four); the first kind of points are the Weierstrass points of weight two and the second kind of weight three. Denoting by f(C) the number of ordinary flexes and by a(C) the number of hyperflexes, on has $$f(C)+2a(C)=24.$$ A very detailed analysis enables the author to obtain both the complete classification of smooth plane quartics with respect to their number of hyperflexes and their configurations (in particular one shows that a(C) can be any integer between 0 and 12 except 10 and 11) and the stratification of $$M_ 3$$. Many comments in the paper refer to the history of the question, as well as to the current research. The paper details the author’s preprint from 1981.
As the author says, similar results has obtained independently E. Lugert [”Weierstraßpunkte kompakter Riemannscher Flächen vom Geschlecht 3” (Thesis, Erlangen-Nürnberg 1981)].
Reviewer: C.Bănică

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H45 Special algebraic curves and curves of low genus 30F10 Compact Riemann surfaces and uniformization 14H10 Families, moduli of curves (algebraic)