Abstract
The aim of this note is to announce some results on the GIT problem for the Hilbert and Chow scheme of curves of degree d and genus g in P^{d-g}, whose full details will appear in a subsequent paper. In particular, we extend the previous results of L. Caporaso up to d>4(2g-2) and we observe that this is sharp. In the range 2(2g-2)<d<7/2(2g-2), we get a complete new description of the GIT quotient. As a corollary, we get a new compactification of the universal Jacobian over the moduli space of pseudo-stable curves.
Highlights
F -theory provides a geometric realization of strongly coupled Type IIB string theory backgrounds
It is motivated by potential applications for building models from string theory, see e.g., [BHV08I, BHV08II]
F -theory will be of the form R3,1 × X, where X is a CalabiYau fourfold admitting an elliptic fibration with a section on a complex threefold B, namely: E G X
Summary
A way to obtain Y6 is to blow up the surfaces ∆1, ∆2 and ∆3 described in Remark 4.2; the strict transform of D6 is a del Pezzo surface of degree 6 on Y6. We consider the subgroup of order 3 of the group of automorphisms isomorphic to Z/6Z defined in Lemma 4.7, with action σ2 := ρ on P5 given by v0 → v5, v1 → v2, v2 → v3, v3 → v1, v4 → v0, v5 → v4. This action leaves Y ′′ invariant, i.e., ρ(Y ′′) = Y ′′. By the MAGMA script FixLocus, it is easy to check that the fixed locus of the action of ρ on Y ′′ consists of 9 isolated points. Three of these points (1 : 1 : 1 : 1 : 1 : 1), (1 : −ζ122 : −ζ122 : −ζ122 : 1 : 1), (1 : ζ122 − 1 : ζ122 − 1 : ζ122 − 1 : 1 : 1) belong to D6
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