The Relative Canonical Ideal of the Kummer–Artin Schreier–Witt family of curves
Annales de l'Institut Fourier, Online first, 29 p.

We study the canonical model of the Kummer–Artin Schreier–Witt flat family of curves over a ring of mixed characteristic. We first prove the relative version of a classical theorem by Petri, then use the model proposed by Bertin–Mézard to construct an explicit generating set for the relative canonical ideal. As a byproduct, we obtain a combinatorial criterion for a set to generate the canonical ideal, applicable to any curve satisfying the assumptions of Petri’s theorem, except for plane quintics and trigonal curves.

Nous étudions le modèle canonique de la famille de courbes plate de Kummer–Artin Schreier–Witt sur un anneau de caractéristique mixte. Nous prouvons d’abord la version relative d’un théorème classique de Petri, puis utilisons le modèle proposé par Bertin–Mézard afin de construire un ensemble de générateurs explicite de l’idéal canonique relatif. De plus, nous obtenons un critère combinatoire pour qu’un ensemble engendre l’idéal canonique, applicable à toute courbe satisfaisante les hypothèses du théorème de Petri, á l’exception des planes quintiques et des courbes trigonales.

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DOI: 10.5802/aif.3543
Classification: 14H10,  14F10
Keywords: Relative curves, Canonical ideal, Reduction and Lifting
Charalambous, Hara 1; Karagiannis, Kostas 1; Kontogeorgis, Aristides 2

1 Department of Mathematics Aristotle University of Thessaloniki School of Sciences 54124, Thessaloniki (Greece)
2 Department of Mathematics National and Kapodistrian University of Athens Panepistimioupolis 1 5784 Athens (Greece)
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Charalambous, Hara; Karagiannis, Kostas; Kontogeorgis, Aristides. The Relative Canonical Ideal of the Kummer–Artin Schreier–Witt family of curves. Annales de l'Institut Fourier, Online first, 29 p.

[1] Bayer, Dave; Eisenbud, David Ribbons and their canonical embeddings, Trans. Amer. Math. Soc., Volume 347 (1995) no. 3, pp. 719-756 | DOI | MR | Zbl

[2] Berkesch, Christine; Schreyer, Frank-Olaf Syzygies, finite length modules, and random curves, Commutative algebra and noncommutative algebraic geometry. Vol. I (Math. Sci. Res. Inst. Publ.), Volume 67, Cambridge Univ. Press, New York, 2015, pp. 25-52 | MR | Zbl

[3] Bertin, José Obstructions locales au relèvement de revêtements galoisiens de courbes lisses, C. R. Acad. Sci. Paris Sér. I Math., Volume 326 (1998) no. 1, pp. 55-58 | DOI | MR | Zbl

[4] 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 | Zbl

[5] Bertin, José; Mézard, Ariane Problem of formation of quotients and base change, Manuscripta Math., Volume 115 (2004) no. 4, pp. 467-487 | DOI | MR | Zbl

[6] Boseck, Helmut Zur Theorie der Weierstrasspunkte, Math. Nachr., Volume 19 (1958), pp. 29-63 | DOI | MR | Zbl

[7] Faltings, Gerd Calculus on arithmetic surfaces, Ann. of Math. (2), Volume 119 (1984) no. 2, pp. 387-424 | DOI | MR | Zbl

[8] Green, Barry; Matignon, Michel Liftings of Galois covers of smooth curves, Compositio Math., Volume 113 (1998) no. 3, pp. 237-272 | MR | Zbl

[9] Griffiths, Phillip; Harris, Joseph Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994, xiv+813 pages (Reprint of the 1978 original) | DOI | MR | Zbl

[10] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York-Heidelberg, 1977, xvi+496 pages | MR

[11] Hartshorne, Robin Deformation theory, Graduate Texts in Mathematics, 257, Springer, New York, 2010, viii+234 pages | DOI | MR | Zbl

[12] Jacobson, Nathan Basic algebra. II, W. H. Freeman and Company, New York, 1989, xviii+686 pages | MR | Zbl

[13] Karanikolopoulos, Sotiris; Kontogeorgis, Aristides Integral representations of cyclic groups acting on relative holomorphic differentials of deformations of curves with automorphisms, Proc. Amer. Math. Soc., Volume 142 (2014) no. 7, pp. 2369-2383 | DOI | MR | Zbl

[14] Nakajima, Shōichi On abelian automorphism groups of algebraic curves, J. London Math. Soc. (2), Volume 36 (1987) no. 1, pp. 23-32 | 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] Obus, Andrew; Wewers, Stefan Cyclic extensions and the local lifting problem, Ann. of Math. (2), Volume 180 (2014) no. 1, pp. 233-284 | DOI | MR | Zbl

[17] Oliveira, Gilvan; Stöhr, Karl-Otto Gorenstein curves with quasi-symmetric Weierstrass semigroups, Geom. Dedicata, Volume 67 (1997) no. 1, pp. 45-63 | MR | Zbl

[18] Oort, Frans Lifting algebraic curves, abelian varieties, and their endomorphisms to characteristic zero, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.), Volume 46, Amer. Math. Soc., Providence, RI, 1987, pp. 165-195 | DOI | MR | Zbl

[19] Pop, Florian The Oort conjecture on lifting covers of curves, Ann. of Math. (2), Volume 180 (2014) no. 1, pp. 285-322 | DOI | MR | Zbl

[20] Saint-Donat, B. On Petri’s analysis of the linear system of quadrics through a canonical curve, Math. Ann., Volume 206 (1973), pp. 157-175 | DOI | MR | Zbl

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

[22] Sekiguchi, T.; Oort, F.; Suwa, N. On the deformation of Artin–Schreier to Kummer, Ann. Sci. École Norm. Sup. (4), Volume 22 (1989) no. 3, pp. 345-375 | DOI | MR | Zbl

[23] Sernesi, Edoardo Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, 334, Springer-Verlag, Berlin, 2006, xii+339 pages | DOI | MR | Zbl

[24] Serre, Jean-Pierre Sur la topologie des variétés algébriques en caractéristique p, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 24-53 | MR | Zbl

[25] Serre, Jean-Pierre Exemples de variétés projectives en caractéristique p non relevables en caractéristique zéro, Proc. Nat. Acad. Sci. U.S.A., Volume 47 (1961), pp. 108-109 | DOI | MR | Zbl

[26] Sturmfels, Bernd Gröbner bases and convex polytopes, University Lecture Series, 8, American Mathematical Society, Providence, RI, 1996, xii+162 pages | DOI | MR | Zbl

[27] Suwa, Noriyuki; Sekiguchi, Tsutomu Théorie de Kummer–Artin–Schreier et applications, Journal de théorie des nombres de Bordeaux, Volume 7 (1995) no. 1, pp. 177-189 http://jtnb.cedram.org/item?id=JTNB_1995__7_1_177_0 | DOI | MR | Zbl

[28] Voisin, Claire Green’s canonical syzygy conjecture for generic curves of odd genus, Compos. Math., Volume 141 (2005) no. 5, pp. 1163-1190 | DOI | MR | Zbl

[29] Witt, Ernst Zyklische Körper und Algebren der Charakteristik p vom Grad p n . Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik p, J. Reine Angew. Math., Volume 176 (1937), pp. 126-140 | DOI | MR | Zbl

[30] Ziegler, Günter M. Lectures on polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, New York, 1995, x+370 pages | DOI | MR | Zbl

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