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Register a new userOn the locus of genus $3$ curves that admit meromorphic differentials with a zero of order $6$ and a pole of order $2$
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The main goal of this article is to compute the class of the divisor of $overline{mathcal{M}}_3$ obtained by taking the closure of the image of $Omegamathcal{M}_3(6;-2)$ by the forgetful map. This is done using Porteous formula and the theory of test curves. For this purpose, we study the locus of meromorphic differentials of the second kind, computing the dimension of the map of these loci to $mathcal{M}_g$ and solving some enumerative problems involving such differentials in low genus. A key tool of the proof is the compactification of strata recently introduced by Bainbridge-Chen-Gendron-Grushevsky-Moller.
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We continue our study of genus 2 curves $C$ that admit a cover $ C to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $L_n$ of the moduli space $M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $L_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $L_5$ as subvarieties of $M_2$ and classify all curves in these loci which have extra automorphisms.
Let the bielliptic locus be the closure in the moduli space of stable curves of the locus of smooth curves that are double covers of genus 1 curves. In this paper we compute the class of the bielliptic locus in bar{M}_3 in terms of a standard basis of the rational Chow group of codimension-2 classes in the moduli space. Our method is to test the class on the hyperelliptic locus: this gives the desired result up to two free parameters, which are then determined by intersecting the locus with two surfaces in bar{M}_3.
Let $K$ be an algebraically closed field of characteristic different from $2$, $g$ a positive integer, $f(x)in K[x]$ a degree $2g+1$ monic polynomial without repeated roots, $C_f: y^2=f(x)$ the corresponding genus g hyperelliptic curve over $K$, and $J$ the jacobian of $C_f$. We identify $C_f$ with the image of its canonical embedding into $J$ (the infinite point of $C_f$ goes to the zero of group law on $J$). It is known (arXiv:1809.03061 [math.AG]) that if $g>1$ then $C_f(K)$ does not contain torsion points, whose order lies between $3$ and $2g$. In this paper we study torsion points of order $2g+1$ on $C_f(K)$. Despite the striking difference between the cases of $g=1$ and $g> 1$, some of our results may be viewed as a generalization of well-known results about points of order $3$ on elliptic curves. E.g., if $p=2g+1$ is a prime that coincides with $char(K)$, then every odd degree genus $g$ hyperelliptic curve contains, at most, two points of order $p$. If $g$ is odd and $f(x)$ has real coefficients, then there are, at most, two real points of order $2g+1$ on $C_f$. If $f(x)$ has rational coefficients and $g<52$, then there are, at most, two rational points of order $2g+1$ on $C_f$. (However, there are exist genus $52$ hyperelliptic curves over the field of rational numbers that have, at least, four rational points of order 105.)
We investigate the geometry of etale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.
We compute the class of the closure of the locus of hyperelliptic curves in the moduli space of stable genus-3 curves in terms of the tautological class $lambda$ and the boundary classes $delta_0$ and $delta_1$. The expression of this class is known, but here we compute it directly, by means of Porteous Formula, without resorting to blowups or test curves.
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