Geometric Invariant Theory for Polarized Curves by Gilberto Bini

By Gilberto Bini

We examine GIT quotients of polarized curves. extra in particular, we research the GIT challenge for the Hilbert and Chow schemes of curves of measure d and genus g in a projective house of size d-g, as d decreases with recognize to g. We end up that the 1st 3 values of d at which the GIT quotients switch are given by means of d=a(2g-2) the place a=2, 3.5, four. We exhibit that, for a>4, L. Caporaso's effects carry real for either Hilbert and Chow semistability. If 3.5<a<4, the Hilbert semistable locus coincides with the Chow semistable locus and it maps to the moduli stack of weakly-pseudo-stable curves. If 2<a<3.5, the Hilbert and Chow semistable loci coincide they usually map to the moduli stack of pseudo-stable curves. We additionally study intimately the severe values a=3.5 and a=4, the place the Hilbert semistable locus is exactly smaller than the Chow semistable locus. As an program, we receive 3 compactications of the common Jacobian over the moduli area of sturdy curves, weakly-pseudo-stable curves and pseudo-stable curves, respectively.

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W /. Part (ii): start by noticing that if every exceptional component E Z which is contracted by is also contracted by 0 , then 0 factors through , so the map exists. Let us now prove that in order for the map to exist, it is also necessary that every exceptional component E Z which is contracted by is also contracted by 0 . By contradiction, assume that exists and that there exists an exceptional component E Z which is contracted by but not by 0 . E 0 / is an exceptional component of Y and that d 0 is properly balanced.

X / D Y , which we call quasiwp-stable (resp. quasi-p-stable, resp. quasi-stable) models of Y . Note that the above operation (ii) cannot occur for quasi-stable curves. (ii)) the bubbling of a node (resp. of a cusp ). Proof We will prove the corollary only for quasi-wp-stable curves. The remaining cases are dealt with in the same way. X /. 11, the wp-stabilization W X ! X / contracts each exceptional component E of X to a node or a cusp according to whether E \ E c consists of two distinct points or one point with multiplicity two.

3] under the assumption that g 3 and then extended to g D 2 with a similar argument by Heyon-Lee in [HL07, Sect. 4]. In what follows, we will show how to adapt the argument of loc. cit. in order to work out in our case. 1 in [HH09], which asserts that given a stable curve C , there is a replacement morphism C W C ! 1 Stable, p-Stable and wp-Stable Curves 21 cusps. The argumentation is local on the nodes connecting each genus-one subcurve meeting the rest of the curve in a single node. Since in a wp-stable curve all elliptic tails are connected to the rest of the curve via a single node, the same argumentation works also in our case with no further modifications.

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