By Gilberto Bini
We examine GIT quotients of polarized curves. extra in particular, we learn the GIT challenge for the Hilbert and Chow schemes of curves of measure d and genus g in a projective house of measurement d-g, as d decreases with recognize to g. We turn out that the 1st 3 values of d at which the GIT quotients switch are given by way of d=a(2g-2) the place a=2, 3.5, four. We convey that, for a>4, L. Caporaso's effects carry precise 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 and so they map to the moduli stack of pseudo-stable curves. We additionally examine intimately the severe values a=3.5 and a=4, the place the Hilbert semistable locus is precisely smaller than the Chow semistable locus. As an program, we receive 3 compactications of the common Jacobian over the moduli house of good curves, weakly-pseudo-stable curves and pseudo-stable curves, respectively.
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Additional resources for Geometric Invariant Theory for Polarized Curves
M/; Symm V _ / ,! m/ Symm V _ dimensional quotients of Symm V _ , which lies naturally in P via the Plücker embedding. V /-equivariant embedding (see [Mum66, Lect. 15]): jm W Hilbd ,! m/; Symm V _ / ,! P. ŒX Pr 7! 1/ and we denote by m WD ss;m Hilbs;m Â Hilbd d Â Hilbd the locus of points that are stable or semistable with respect to m , respectively. If ŒX Pr 2 Hilbs;m Pr 2 Hilbss;m Pr is m-Hilbert d (resp. ŒX d ), we say that ŒX stable (resp. semistable). 3(i)]. In particular, Hilbs;m are d and Hilbd constant for m 0.
V / (see [Gie82, pp. V /, and conversely). Let W Gm ! t/ xi D t wi xi with wi 2 Z: 48 4 Preliminaries on GIT The total weight of is by definition w. B/ D r X ˇi wi : i D0 For any m function M as in Sect. m//. m/ coincide with the filtered Hilbert function of [HH13, Def. 15]. m/ when there is no danger of confusion. The Hilbert-Mumford numerical criterion for m-th Hilbert (semi)stability translates into the following (see [Gie82, p. 10] and also [HM98, Prop. 23]). 2 (Numerical Criterion for m-Hilbert (Semi)stability ) Let m M as before.
Clearly, X is a stable and p-stable curve 1/ of genus g, so that Xexc D ;. L/. Therefore, L is properly balanced but not strictly balanced. Finally, let us now prove the implication (iv))(i). e. d; g 1/ ¤ 1, and we will construct a curve XQ of genus g which is both quasi-stable and quasi-p-stable (hence in particular Q on XQ of degree d which is strictly quasi-wp-stable) together with a line bundle L Q balanced but not stably balanced. Indeed, let X be the curve obtained from the curve X constructed above (in Case 1 and in Case 2) by bubbling all the nodes.
Geometric Invariant Theory for Polarized Curves by Gilberto Bini