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Weierstrass cycles and tautological rings in various moduli spaces of algebraic curves. (English) Zbl 1345.14042

A numerical semigroup \(H\) is a subsemigroup of the non-negative integers \(\mathbb{N}_0\) such that the greatest common divisor of the elements of \(H\) is \(1\), and \(\mathbb{N}_0 \setminus H\) is finite. The elements of \(\mathbb{N}_0 \setminus H\) are called gaps, and their number \(g(H)\), the genus of \(H\).
Now, let \(C\) be a smooth complex irreducible projective curve of genus \(g\), and \(p \in C\). Let \(H_p\) be the set consisting of \(0\) and the integers \(n\) such that there exists a function on \(C\) holomorphic everywhere except at \(p\), and having a pole of order \(n\) at \(p\). Then, \(H_p\) is a numerical semigroup of genus \(g\), called the Weierstrass semigroup of \(p\).
Let \(H\) be a numerical semigroup of genus \(g\), and denote by \(\mathcal{M}_H\) the space of points \((C,p)\) such that \(p\) has Weierstrass semigroup \(H\). \(\mathcal{M}_H\) is a subspace of the moduli space \(\mathcal{M}_{g,1}\) of complex smooth pointed curves of genus \(g\). The closure of \(\mathcal{M}_H\) is called a Weierstrass cycle of semigroup \(H\), and denoted by \(W_H\).
The present paper is devoted to study the dimension of Weierstrass cycles. The first two authors obtained in the preprint [“Weierstrass cycles in moduli spaces and the Krivecher map”, Preprint, arXiv:1207.0530] a lower estimate for that dimension. Now, the case of low genus, namely \(g \leq 6\), is considered. It is proved that the estimate is precise for \(g \leq 5\), whilst for \(g=6\) the error is at most \(1\). Also, the cohomology classes of \(W_H\) are obtained for \(g \leq 6\).

MSC:

14H55 Riemann surfaces; Weierstrass points; gap sequences
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14C25 Algebraic cycles
14H10 Families, moduli of curves (algebraic)
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References:

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