## 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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