no. G6
Géométrie algébrique
The connectedness of degeneracy loci in positive characteristic
Lodh, Rémi1
1 Springer-Verlag, Tiergartenstr. 17, 69121 Heidelberg, Germany
Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 959-964

A well-known result of Fulton–Lazarsfeld ensures the connectedness of degeneracy loci under an ampleness condition. We extend it to positive characteristic, along with the variants for degeneracy loci of symmetric and alternating maps of even rank, due to Tu in characteristic zero. The proof uses the explicit determination of the top étale cohomology group of an algebraic variety, a result communicated by Esnault.

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DOI : 10.5802/crmath.448
Classification : 14F20, 14N05, 14J60, 14F06, 14M12, 14F45, 14E15, 14G17

Lodh, Rémi 1

1 Springer-Verlag, Tiergartenstr. 17, 69121 Heidelberg, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Lodh, Rémi. The connectedness of degeneracy loci in positive characteristic. Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 959-964. doi: 10.5802/crmath.448

[1] Artin, Emil Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics, 3, Interscience Publishers, 1957

[2] Deligne, Pierre Exposé XVIII: La formule de dualité globale, Théorie des topos et cohomologie étale des schémas (SGA 4), tome 3 (Artin, Michel; Grothendieck, Alexander; Verdier, Jean-Louis; Deligne, Pierre; Saint-Donat, Bernard, eds.) (Lecture Notes in Mathematics), Volume 305, Springer, 1973 | DOI | Zbl

[3] Flenner, Hubert; Ulrich, Bernd Codimension and connectedness of degeneracy loci over local rings, Math. Z., Volume 286 (2017) no. 1-2, pp. 723-740 | DOI | MR | Zbl

[4] Fulton, William; Lazarsfeld, Robert On the connectedness of degeneracy loci and special divisors, Acta Math., Volume 146 (1981), pp. 271-283 | DOI | MR | Zbl

[5] Fulton, William; Lazarsfeld, Robert Positive polynomials for ample vector bundles, Ann. Math., Volume 118 (1983), pp. 35-60 | DOI | MR | Zbl

[6] Grothendieck, Alexander Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie), Publ. Math., Inst. Hautes Étud. Sci., Volume 32 (1967), pp. 5-361 (written in collaboration with J. Dieudonné) | Numdam | Zbl

[7] Grothendieck, Alexander; Dieudonné, Jean A. Éléments de géométrie algébrique I, Grundlehren der Mathematischen Wissenschaften, 166, Springer, 1971

[8] Hartshorne, Robin Ample vector bundles, Publ. Math., Inst. Hautes Étud. Sci., Volume 29 (1966), pp. 63-94 | Numdam | Zbl

[9] Illusie, Luc; Temkin, Michael Exposé X. Gabber’s modification theorem (log smooth case), Travaux de Gabber sur l’uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l’École Polytechnique 2006–2008 (Illusie, Luc; Laszlo, Yves; Orgogozo, Fabrice, eds.) (Astérisque), Société Mathématique de France, 2014 no. 363-364, pp. 167-212 | Zbl

[10] de Jong, Aise J. Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci., Volume 83 (1996), pp. 51-93 | Zbl | Numdam

[11] Lazarsfeld, Robert Positivity in Algebraic Geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer, 2004

[12] Scharlau, Winfred Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften, 270, Springer, 1985 | DOI

[13] Tin, Siu-Kei Numerically positive polynomials for k-ample vector bundles, Math. Ann., Volume 294 (1992) no. 4, pp. 579-590 | MR | Zbl

[14] Tu, Loring W. The connectedness of symmetric and skew-symmetric degeneracy loci: even ranks, Trans. Am. Math. Soc., Volume 313 (1989) no. 1, pp. 381-392 | MR | Zbl

[15] Tu, Loring W. The connectedness of degeneracy loci, Topics in algebra. Part 2: Commutative rings and algebraic groups (Banach Center Publications), Volume 26, PWN-Polish Scientific Publishers, 1990 no. 2, pp. 235-248 | MR | Zbl

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