Hostname: page-component-5b777bbd6c-5mwv9 Total loading time: 0 Render date: 2025-06-20T05:34:25.835Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert–Mumford criterion for nodal curves

Part of: Curves Complex manifolds

Published online by Cambridge University Press:  10 June 2015

Jun Li  and
Xiaowei Wang
Jun Li
Affiliation:
Stanford University, Stanford, CA 94305, USA email jli@stanford.edu
Xiaowei Wang
Affiliation:
The Chinese University of Hong Kong, Hong Kong and Rutgers University, Newark, NJ 07102, USA email xiaowwan@rutgers.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove by the Hilbert–Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves and of Hassett on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solves a question raised by Mumford and Gieseker, to prove the Chow asymptotic stability of stable nodal curves by the Hilbert–Mumford criterion.

Keywords

geometric invariant theoryasymptotic stabilityChow stabilityChow weightK-stabilityDonaldson–Futaki invariantmoduli of curvesone parameter subgroupsHilbert–Mumford criterion

MSC classification

Primary: 14H10: Families, moduli (algebraic)
Secondary: 32Q20: Kähler-Einstein manifolds
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 11 , November 2015 , pp. 2076 - 2130
Copyright
© The Authors 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Baldwin, E. and Swinarski, D., A geometric invariant theory construction of moduli spaces of stable maps, Int. Math. Res. Pap. IMRP (2008), Art. ID rpn 004.Google Scholar
Bini, G., Felici, F., Melo, M. and Viviani, F., Geometric invariant theory for polarized curves, Lecture Notes in Mathematics, vol. 2122 (Springer, Cham, 2014).CrossRefGoogle Scholar
Caporaso, L., A compactification of the universal Picard variety over the moduli space of stable curves, J. Amer. Math. Soc. 7 (1994), 589–660.CrossRefGoogle Scholar
Donaldson, S. K., Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289–349.CrossRefGoogle Scholar
Eisenbud, D., Commutative algebra, in With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995).Google Scholar
Gieseker, D., On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2) 106 (1977), 45–60.CrossRefGoogle Scholar
Gieseker, D., Lectures on moduli of curves, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69 (Springer, New York, 1982).Google Scholar
Gieseker, D., Geometric invariant theory and the moduli of bundles, in Gauge theory and the topology of four-manifolds (Park City, UT, 1994), IAS/Park City Mathematics Series, vol. 4 (American Mathematical Society, Providence, RI, 1998).Google Scholar
Gieseker, D. and Morrison, I., Hilbert stability of rank-two bundles on curves, J. Differential Geom. 19 (1984), 1–29.CrossRefGoogle Scholar
Hassett, B., Moduli spaces of weighted pointed stable curves, Adv. Math. 173 (2003), 316–352.CrossRefGoogle Scholar
Harris, J. and Morrison, I., Moduli of curves, Graduate Texts in Mathematics, vol. 187 (Springer, New York, 1998).Google Scholar
Kantor, J. and Khovanskii, A., Une application du théorème de Riemann–Roch combinatoire au polynôme d’Ehrhart des polytopes entiers de ℝd, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 501–507.Google Scholar
Lazarsfeld, R., Positivity in algebraic geometry. II., in Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3 (Springer, Berlin, 2004).Google Scholar
Morrison, I., GIT constructions of moduli spaces of stable curves and maps, in Surveys in differential geometry, Vol. XIV, Geometry of Riemann surfaces and their moduli spaces, Surveys in Differential Geometry, vol. 14 (International Press, Somerville, MA, 2009), 315–369; arXiv:0810.2340.Google Scholar
Mumford, D., Stability of projective varieties, Enseign. Math. (2) 23 (1977), 39–110.Google Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, third edition (Springer, Berlin, 1994).CrossRefGoogle Scholar
Odaka, Y., On the moduli of Kähler–Einstein Fano manifolds, Preprint (2012),arXiv:1211.4833.Google Scholar
Odaka, Y., The GIT-stability of polarized varieties via discrepancy, Ann. of Math. (2) 177 (2013), 645–661; arXiv:0807.1716.CrossRefGoogle Scholar
Odaka, Y., A generalization of Ross–Thomas’ slope theory, Osaka J. Math. 50 (2013), 171–185; arXiv:0910.1794.Google Scholar
Odaka, Y. and Sun, S., Testing log K-stability by blowing up formalism, Ann. Fac. Sci. Toulouse Math., to appear, arXiv:1112.1353.Google Scholar
Paul, S. and Tian, G., CM stability and the generalized Futaki invariant I, Preprint (2006),arXiv:math/0605278 [math.AG].Google Scholar
Ross, J. and Thomas, R., A study of the Hilbert–Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), 201–255.CrossRefGoogle Scholar
Schubert, D., A new compactification of the moduli space of curves, Compositio Math. 78 (1991), 297–313.Google Scholar
Stoppa, J., K-stability of constant scalar curvature Kähler manifolds, Adv. Math. 221 (2009), 1397–1408.CrossRefGoogle Scholar
Stoppa, J., A note on the definition of $K$-stability, Preprint (2011), arXiv:1111.5826.Google Scholar
Swinarski, D., GIT stability of weighted pointed curves, Trans. Amer. Math. Soc. 364 (2012), 1737–1770.CrossRefGoogle Scholar
Wang, X., Height and GIT weight, Math. Res. Lett. 19 (2012), 906–926.CrossRefGoogle Scholar
Wang, X. and Xu, C., Nonexistence of asymptotic GIT compactification, Duke Math. J. 163 (2014), 2217–2241.CrossRefGoogle Scholar