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Footnotes to a paper of Beniamino Segre. (The number of $$g^1_d$$’s on a general $$d$$-gonal curve, and the unirationality of the Hurwitz spaces of 4-gonal and 5-gonal curves). (English) Zbl 0454.14023
Summary: This work was inspired by B. Segre’ s paper “Sui moduli delle curve poligonali e sopra un complemento al teorema di esistenza di Riemann” [Math. Ann. 100, 537–551 (1928; JFM 54.0685.01)]. Segre proves that a general $$d$$-gonal curve, $$2\le d<g/2 + 1$$, has finitely many $$g_d^1$$’s, and moreover that a general $$d$$-gonal curve, $$d\ge 3$$ can be realized as a plane curve of degree $$n$$, for any $$n\ge (g +d +2)/2$$, having as its only singularities an $$(n-d)$$-fold point $$p$$ and nodes, in such a way that the given $$g_d^1$$ is obtained by projecting from $$p$$.
In the first part of this paper the authors show that a general $$d$$-gonal curve of genus $$g$$ possesses no $$g_h^1$$, $$2\le h\le g/2 + 1$$, other than those composed with the given $$g_d^1$$.
In the second part of the paper it is shown that, given general points $$p_1, \ldots, p_\delta$$ in $$\mathbb P^2$$ and an integer $$n$$ such that $$3\delta\le \binom{n+2}{2} - 1$$, $$\delta\le \binom{n - 1}{2}$$, there exists an irreducible plane curve of degree $$n$$ having nodes at $$p_1, \ldots, p_\delta$$ as its only singularities, with the exception of the case $$\delta = 9$$, $$n = 6$$, when the only curve of degree $$n$$ passing doubly through $$p_1, \ldots, p_\delta$$ is a cubic counted twice.
This result is also generalized to arbitrary rational surfaces and applied to show that the Hurwitz spaces of 4-gonal and 5-gonal curves of any genus and of 6-gonal curves of genus $$g\le 10$$ or $$g= 12$$ are unirational.
Reviewer: Enrico Arbarello

##### MSC:
 14N10 Enumerative problems (combinatorial problems) in algebraic geometry 14M20 Rational and unirational varieties 14H45 Special algebraic curves and curves of low genus
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##### References:
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