## 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)].

### MSC:

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

### Keywords:

Weierstrass semigroups; moduli space of pointed curves

### Citations:

Zbl 0606.14014; Zbl 0272.14003; Zbl 1273.14069; Zbl 0771.14009
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