Quadratic Differentials and Equivariant Deformation Theory of Curves

Köck, Bernhard; Kontogeorgis, Aristides

Annales de l'Institut Fourier, Tome 62 (2012) no. 3, pp. 1015-1043.

Soit G un $p$-groupe fini agissant sur une courbe lisse projective $X$ sur un corps algébriquement clos $k$ de caractéristique $p$. Alors la dimension de l’espace tangent du foncteur de déformations équivariantes associé est égal à la dimension de l’espace de co-invariants de $G$ agissant sur l’espace $V$ de différentielles holomorphes quadratiques globales sur $X$. On applique des résultats connus sur la structure de module de Galois des espaces Riemann-Roch pour calculer cette dimension dans le cas où $G$ est cyclique ou dans le cas où l’action de $G$ sur $X$ est faiblement ramifiée. De plus, on détermine certaines sous-représentations de $V$, qui s’appellent rang-$p$ représentations.

Given a finite $p$-group $G$ acting on a smooth projective curve $X$ over an algebraically closed field $k$ of characteristic $p$, the dimension of the tangent space of the associated equivariant deformation functor is equal to the dimension of the space of coinvariants of $G$ acting on the space $V$ of global holomorphic quadratic differentials on $X$. We apply known results about the Galois module structure of Riemann-Roch spaces to compute this dimension when $G$ is cyclic or when the action of $G$ on $X$ is weakly ramified. Moreover we determine certain subrepresentations of $V$, called $p$-rank representations.

Classification : 14H30, 14D15, 14F10, 11R32
Keywords: quadratic differentials, tangent space, equivariant deformation functor, Galois modules, Riemann-Roch spaces, weakly ramified, $p$-rank representation
Mot clés : différentiels quadratiques, espace tangent, foncteur de déformations équivariantes, modules de Galois, espaces Riemann-Roch, faiblement ramifiée, rang-$p$ réprésentations

[1] Bertin, José; Mézard, Ariane Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math., Volume 141 (2000) no. 1, pp. 195-238 | DOI | MR

[2] Borne, Niels A relative Shafarevich theorem, Math. Z., Volume 248 (2004) no. 2, pp. 351-367 | DOI | MR

[3] Borne, Niels Cohomology of $G$-sheaves in positive characteristic, Adv. Math., Volume 201 (2006) no. 2, pp. 454-515 | DOI | MR

[4] Cornelissen, Gunther; Kato, Fumiharu Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic, Duke Math. J., Volume 116 (2003) no. 3, pp. 431-470 | DOI | MR

[5] Cornelissen, Gunther; Kato, Fumiharu Zur Entartung schwach verzweigter Gruppenoperationen auf Kurven, J. Reine Angew. Math., Volume 589 (2005), pp. 201-236 | DOI | MR

[6] Farkas, Hershel M.; Kra, Irwin Riemann surfaces, Graduate Texts in Mathematics, 71, Springer-Verlag, New York, 1980 | MR | Zbl

[7] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, Springer-Verlag, New York, 1977 no. 52 | MR | Zbl

[8] Karanikolopoulos, Sotiris On holomorphic polydifferentials in positive characteristic, to appear in Math. Nachr., 25pp, 2010 (arXiv:0905.1196v2)

[9] Köck, Bernhard Galois structure of Zariski cohomology for weakly ramified covers of curves, Amer. J. Math., Volume 126 (2004) no. 5, pp. 1085-1107 http://muse.jhu.edu/journals/american_journal_of_mathematics/v126/126.5k"ock.pdf | DOI | MR

[10] Kontogeorgis, Aristides On the tangent space of the deformation functor of curves with automorphisms, Algebra Number Theory, Volume 1 (2007) no. 2, pp. 119-161 | DOI | MR

[11] Kontogeorgis, Aristides Polydifferentials and the deformation functor of curves with automorphisms, J. Pure Appl. Algebra, Volume 210 (2007) no. 2, pp. 551-558 | DOI | MR

[12] Kontogeorgis, Aristides The ramification sequence for a fixed point of an automorphism of a curve and the Weierstrass gap sequence, Math. Z., Volume 259 (2008) no. 3, pp. 471-479 | DOI | MR

[13] Nakajima, Shōichi Equivariant form of the Deuring-Šafarevič formula for Hasse-Witt invariants, Math. Z., Volume 190 (1985) no. 4, pp. 559-566 | DOI | MR | Zbl

[14] Nakajima, Shōichi Action of an automorphism of order $p$ on cohomology groups of an algebraic curve, J. Pure Appl. Algebra, Volume 42 (1986) no. 1, pp. 85-94 | DOI | MR | Zbl

[15] Nakajima, Shōichi $p$-ranks and automorphism groups of algebraic curves, Trans. Amer. Math. Soc., Volume 303 (1987) no. 2, pp. 595-607 | DOI | MR | Zbl

[16] Schlessinger, Michael Functors of Artin rings, Trans. Amer. Math. Soc., Volume 130 (1968), pp. 208-222 | DOI | MR | Zbl

[17] Serre, Jean-Pierre Linear representations of finite groups, Graduate Texts in Mathematics, Translated from the second French edition by Leonard L. Scott, 42, Springer-Verlag, New York, 1977 | MR | Zbl

[18] Serre, Jean-Pierre Local fields, Graduate Texts in Mathematics, Translated from the French by Marvin Jay Greenberg, 67, Springer-Verlag, New York, 1979 | MR | Zbl

[19] Silverman, Joseph H. The arithmetic of elliptic curves, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009 | DOI | MR | Zbl

[20] Stalder, Nicolas On $p$-rank representations, J. Algebra, Volume 280 (2004) no. 2, pp. 825-841 | DOI | MR

[21] Stichtenoth, Henning Algebraic function fields and codes, Graduate Texts in Mathematics, 254, Springer-Verlag, Berlin, 2009 | MR | Zbl

[22] Subrao, Doré The $p$-rank of Artin-Schreier curves, Manuscripta Math., Volume 16 (1975) no. 2, pp. 169-193 | DOI | MR | Zbl

[23] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994 | MR | Zbl