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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**Example text**

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.