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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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