American Institute of Mathematical Sciences

2012, 19: 33-40. doi: 10.3934/era.2012.19.33

On GIT quotients of Hilbert and Chow schemes of curves

 1 Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy 2 Departamento de Matemática, Universidade de Coimbra, Largo D. Dinis, Apartado 3008, 3001 Coimbra, Portugal 3 Dipartimento di Matematica, Università Roma Tre, Largo S. Leonardo Murialdo 1, 00146 Roma, Italy

Received  September 2011 Revised  January 2012 Published  February 2012

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 the projective space of dimension $d-g$, whose full details will appear in [6]. 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 $2(2g-2) < d < \frac{7}{2} (2g-2)$, 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.
Citation: Gilberto Bini, Margarida Melo, Filippo Viviani. On GIT quotients of Hilbert and Chow schemes of curves. Electronic Research Announcements, 2012, 19: 33-40. doi: 10.3934/era.2012.19.33
References:
 [1] J. Alper, Adequate moduli spaces and geometrically reductive group schemes, preprint, arXiv:1005.2398. [2] J. Alper and D. Hyeon, GIT construction of log canonical models of $\barM_g$, preprint, arXiv:1109.2173. [3] J. Alper, D. Smyth and M. Fedorchuck, Finite Hilbert stability of (bi)canonical curves, preprint, arXiv:1109.4986. [4] J. Alper, D. Smyth and M. Fedorchuck, Finite Hilbert stability of canonical curves, II. The even-genus case, preprint, arXiv:1110.5960. [5] J. Alper, D. Smyth and F. van der Wick, Weakly proper moduli stacks of curves, preprint, arXiv:1012.0538. [6] G. Bini, M. Melo and F. Viviani, GIT for polarized curves, preprint, arXiv:1109.6908v2. [7] L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc., 7 (1994), 589-660. doi: 10.1090/S0894-0347-1994-1254134-8. [8] M. Fedorchuk and D. Jensen, Stability of 2nd Hilbert points of canonical curves, preprint, arXiv:1111.5339. [9] M. Fedorchuk and D. I. Smyth, Alternate compactifications of moduli space of curves, to appear in "Handbook of Moduli" (eds. G. Farkas and I. Morrison), arXiv:1012.0329. [10] F. Felici, GIT for curves of low degree, in progress. [11] D. Gieseker, "Lectures on Moduli of Curves," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 69, Tata Institute of Fundamental Research, Bombay, 1982. [12] J. Harris and I. Morrison, "Moduli of Curves," Graduate Text in Mathematics, 187, Springer-Verlag, New York, 1998. [13] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: First divisorial contraction, Trans. Amer. Math. Soc., 361 (2009), 4471-4489. doi: 10.1090/S0002-9947-09-04819-3. [14] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: The first flip, preprint arXiv:0806.3444. [15] D. Hyeon and Y. Lee, Stability of tri-canonical curves of genus two, Math. Ann., 337 (2007), 479-488. doi: 10.1007/s00208-006-0046-2. [16] D. Hyeon and I. Morrison, Stability of tails and 4-canonical models, Math. Res. Lett., 17 (2010), 721-729. [17] J. Li and X. Wang, Hilbert-Mumford criterion for nodal curves, preprint, arXiv:1108.1727v1. [18] I. Morrison, GIT constructions of moduli spaces of stable curves and maps, in "Geometry of Riemann surfaces and their Moduli Spaces" (eds. L. Ji, et al.), Surveys in Differential Geometry 14, International Press, Somerville, MA, (2010), 315-369. [19] D. Mumford, "Lectures on Curves on an Algebraic Surface," Annals of Mathematics Studies, 59, Princeton University Press, Princeton, N.J., 1966. [20] D. Mumford, Stability of projective varieties, Enseignement Math. (2), 23 (1977), 39-110. [21] D. Schubert, A new compactification of the moduli space of curves, Compositio Math., 78 (1991), 297-313.

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References:
 [1] J. Alper, Adequate moduli spaces and geometrically reductive group schemes, preprint, arXiv:1005.2398. [2] J. Alper and D. Hyeon, GIT construction of log canonical models of $\barM_g$, preprint, arXiv:1109.2173. [3] J. Alper, D. Smyth and M. Fedorchuck, Finite Hilbert stability of (bi)canonical curves, preprint, arXiv:1109.4986. [4] J. Alper, D. Smyth and M. Fedorchuck, Finite Hilbert stability of canonical curves, II. The even-genus case, preprint, arXiv:1110.5960. [5] J. Alper, D. Smyth and F. van der Wick, Weakly proper moduli stacks of curves, preprint, arXiv:1012.0538. [6] G. Bini, M. Melo and F. Viviani, GIT for polarized curves, preprint, arXiv:1109.6908v2. [7] L. Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc., 7 (1994), 589-660. doi: 10.1090/S0894-0347-1994-1254134-8. [8] M. Fedorchuk and D. Jensen, Stability of 2nd Hilbert points of canonical curves, preprint, arXiv:1111.5339. [9] M. Fedorchuk and D. I. Smyth, Alternate compactifications of moduli space of curves, to appear in "Handbook of Moduli" (eds. G. Farkas and I. Morrison), arXiv:1012.0329. [10] F. Felici, GIT for curves of low degree, in progress. [11] D. Gieseker, "Lectures on Moduli of Curves," Tata Institute of Fundamental Research Lectures on Mathematics and Physics, 69, Tata Institute of Fundamental Research, Bombay, 1982. [12] J. Harris and I. Morrison, "Moduli of Curves," Graduate Text in Mathematics, 187, Springer-Verlag, New York, 1998. [13] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: First divisorial contraction, Trans. Amer. Math. Soc., 361 (2009), 4471-4489. doi: 10.1090/S0002-9947-09-04819-3. [14] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: The first flip, preprint arXiv:0806.3444. [15] D. Hyeon and Y. Lee, Stability of tri-canonical curves of genus two, Math. Ann., 337 (2007), 479-488. doi: 10.1007/s00208-006-0046-2. [16] D. Hyeon and I. Morrison, Stability of tails and 4-canonical models, Math. Res. Lett., 17 (2010), 721-729. [17] J. Li and X. Wang, Hilbert-Mumford criterion for nodal curves, preprint, arXiv:1108.1727v1. [18] I. Morrison, GIT constructions of moduli spaces of stable curves and maps, in "Geometry of Riemann surfaces and their Moduli Spaces" (eds. L. Ji, et al.), Surveys in Differential Geometry 14, International Press, Somerville, MA, (2010), 315-369. [19] D. Mumford, "Lectures on Curves on an Algebraic Surface," Annals of Mathematics Studies, 59, Princeton University Press, Princeton, N.J., 1966. [20] D. Mumford, Stability of projective varieties, Enseignement Math. (2), 23 (1977), 39-110. [21] D. Schubert, A new compactification of the moduli space of curves, Compositio Math., 78 (1991), 297-313.
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