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Stratification of the moduli space of four-gonal curves. (English) Zbl 1304.14031
A \(\gamma\)-gonal curve is a smooth irreducible projective curve that admits a degree \(\gamma\) branched cover of \(\mathbb P^1\). The locus of \(\gamma\)-gonal curves \(\mathcal M_{g, \gamma}\) provides an important series of subvarieties of the moduli space of genus \(g\) curves \(\mathcal M_g\). (Caution: here the notation \(\mathcal M_{g, \gamma}\) does not stand for the moduli space of pointed curves.) For instance, if \(d=2\), \(\mathcal M_{g, 2}\) is the locus of hyperelliptic curves. For \(d=3\), the canonical model of a trigonal curve \(X\) lies in a scroll surface, which is spanned by the trigonal divisors on \(X\). The (un)balanced type of the scroll surface defines the so-called Maroni invariant of \(X\). In particular, loci of trigonal curves with the same Maroni invariant yield a stratification of \(\mathcal M_{g, 3}\).
In the paper under review, the authors study the locus of four-gonal curves \(\mathcal M_{g, 4}\). Analogous to the case of trigonal curves, they show that for a four-gonal curve \(X\), there (almost always uniquely) exists a surface, ruled by conics, containing the canonical model of \(X\). By analyzing the geometry of such a surface, the authors are able to extract several discrete invariants associated to \(X\). They further study the relation of those invariants as well as their geometric meanings and possible ranges. Finally, the authors apply the invariants to describe a stratification of \(\mathcal M_{g, 4}\). The existence of such a surface has also been obtained, using a different method, by F.-O. Schreyer [Math. Ann. 275, 105–137 (1986; Zbl 0578.14002)].

MSC:
14H10 Families, moduli of curves (algebraic)
14N05 Projective techniques in algebraic geometry
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