Upper bounds for the dimension of moduli spaces of curves with symmetric Weierstrass semigroups. (English) Zbl 1288.14016

Let \(V:={\mathcal M}_H\) denote the moduli space of smooth complete integral pointed algebraic curves with a prescribed Weierstrass semigroup \(H\) of genus \(g\). The subject matter here is concerning the dimension \(d(V)\) of \(V\). By a result of D. Eisenbud and J. Harris \(d(V)\geq 3g-2-w(H)\), where \(w(H)\) is the weight of \(H\) [Invent. Math. 87, 495–515 (1987; Zbl 0606.14014)], this bound being attained if either \(w(H)\leq g/2\), or \(H\) is primitive with \(w(H)<g\). On the other hand, by a result of P. Deligne \(d(V)\leq 2g-2+\lambda(H)\), where \(\lambda(H)\) is the number of gaps \(\ell\) of \(H\) such that \(\ell+h\in H\) for each \(h\in H\setminus\{0\}\) [in: Sem. Geom. algebrique Bois–Marie 1967–1969, SGA 7 II, Lect. Notes Math. 340, Expose X, 1–38 (1973; Zbl 0272.14003)]. In particular, if \(H\) is symmetric \(d(V)\leq 2g-1\) and equality holds whenever the gap sequence of \(H\) is either \(\{1,\ldots,g-1,2g-1\}\), or \(\{1,\ldots,g-2,g,2g-1\}\), or \(\{1,3,5,\ldots,2g-1\}\) [E. M. Bullock, Trans. Am. Math. Soc. 365, No. 1, 99–122 (2013; Zbl 1273.14069)].
In this paper the authors give a method to improve on Deligne’s bound above whenever \(H\) is a symmetric semigroup. Indeed, the point is to construct an appropriate compactification for \(V\) and they do that by taking into consideration a previous construction due to K.-O. Stöhr [J. Reine Angew. Math. 441, 189–213 (1993; Zbl 0771.14009)].


14H10 Families, moduli of curves (algebraic)
14H55 Riemann surfaces; Weierstrass points; gap sequences
14Q05 Computational aspects of algebraic curves
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