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Register a new userExtended Torelli map to the Igusa blowup in genus 6, 7, and 8
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It was conjectured in cite{Namikawa_ExtendedTorelli} that the Torelli map $M_gto A_g$ associating to a curve its jacobian extends to a regular map from the Deligne-Mumford moduli space of stable curves $bar{M}_g$ to the (normalization of the) Igusa blowup $bar{A}_g^{rm cent}$. A counterexample in genus $g=9$ was found in cite{AlexeevBrunyate}. Here, we prove that the extended map is regular for all $gle8$, thus completely solving the problem in every genus.
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We prove that the ramified Prym map $mathcal P_{g, r}$ which sends a covering $pi:Dlongrightarrow C$ ramified in $r$ points to the Prym variety $P(pi):=text{Ker}(text{Nm}_{pi})$ is an embedding for all $rge 6$ and for all $g(C)>0$. Moreover, by studying the restriction to the locus of coverings of hyperelliptic curves, we show that $mathcal P_{g, 2}$ and $mathcal P_{g, 4}$ have positive dimensional fibers.
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.
The Prym map assigns to each covering of curves a polarized abelian variety. In the case of unramified cyclic covers of curves of genus two, we show that the Prym map is ramified precisely on the locus of bielliptic covers. The key observation is that we can naturally associate to such a cover an abelian surface with a cyclic polarization, and then the codifferential of the Prym map can be interpreted in terms of multiplication of sections on the abelian surface. Furthermore, we prove that a genus two cyclic cover of degree at least seven is never hyperelliptic.
We study the Prym varieties arising from etale cyclic coverings of degree 7 over a curve of genus 2. These Prym varieties are products of Jacobians JY x JY of genus 3 curves Y with polarization type D=(1,1,1,1,1,7). We describe the fibers of the Prym map between the moduli space of such coverings and the moduli space of abelian sixfolds with polarization type D, admitting an automorphism of order 7.
Candidate counterterms break Noether-Gaillard-Zumino E_{7(7)} current conservation in N=8 supergravity in four dimensions. Bossard and Nicolai proposed a scheme for deforming the subsector involving vector fields in a Lorentz covariant manner, so as to restore duality. They argued that there must exist an extension of this deformation to the full theory that preserves supersymmetry. We show that it is not possible to deform the maximal supergravity to restore E_{7(7)} duality, while maintaining both general covariance and N=8 supersymmetry, as was proposed. Deformation of N=8 supergravity requires higher spins and multiple gravitons, which presents a concrete obstacle to this proposal.
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