Varieties with ample Frobenius-trace kernel
Annales Henri Lebesgue, Volume 8 (2025), pp. 721-767

Metadata

Keywords Cokernel of Frobenius, Cartier operators, Frobenius traces, Kunz theorem, Mori–Hartshorne theorem

Abstract

In the search of a projective analog of Kunz’s theorem and a Frobenius-theoretic analog of Mori–Hartshorne’s theorem, we investigate the positivity of the kernel of the Frobenius trace (equivalently, the negativity of the cokernel of the Frobenius endomorphism) on a smooth projective variety over an algebraically closed field of positive characteristic. For instance, such a kernel is ample for projective spaces. Conversely, we show that for curves, surfaces, and threefolds the Frobenius trace kernel is ample only for Fano varieties of Picard rank $1$.


References

[Ach12] Achinger, Piotr Frobenius push-forwards on quadrics, Commun. Algebra, Volume 40 (2012) no. 8, pp. 2732-2748 | DOI | MR | Zbl

[Ach15] Achinger, Piotr A characterization of toric varieties in characteristic p, Int. Math. Res. Not., Volume 2015 (2015) no. 16, pp. 6879-6892 | DOI | MR | Zbl

[Add09] Addington, Nicolas Spinor sheaves and complete intersections of quadrics, Ph. D. Thesis, University of Wisconsin – Madison (2009) (https://pages.uoregon.edu/adding/theses/phd_thesis.pdf) | MR

[Add11] Addington, Nicolas Spinor sheaves on singular quadrics, Proc. Am. Math. Soc., Volume 139 (2011) no. 11, pp. 3867-3879 | MR | Zbl | DOI

[BH93] Bruns, Winfried; Herzog, Jürgen Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, 39, Cambridge University Press, 1993 | MR | Zbl

[BK86] Bloch, Spencer; Kato, Kazuya p-adic etale cohomology, Publ. Math., Inst. Hautes Étud. Sci., Volume 63 (1986), pp. 107-152 | DOI | MR | Zbl | Numdam

[BK05] Brion, Michel; Kumar, Shrawan Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, 231, Birkhäuser, 2005 | MR | Zbl | DOI

[Car57] Cartier, Pierre Une nouvelle opération sur les formes différentielles, C. R. Acad. Sci. Paris, Volume 244 (1957), pp. 426-428 | MR | Zbl

[EH16] Eisenbud, David; Harris, Joe 3264 and all that. A second course in algebraic geometry, Cambridge University Press, 2016 | DOI | MR | Zbl

[ES19] Ejiri, Sho; Sannai, Akiyoshi A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf. II, Int. Math. Res. Not., Volume 2019 (2019) no. 19, pp. 5975-5988 | DOI | MR | Zbl

[EV92] Esnault, Hélène; Viehweg, Eckart Lectures on vanishing theorems, DMV Seminar, 20, Birkhäuser, 1992 | MR | Zbl | DOI

[FS20] Fanelli, Andrea; Schröer, Stefan Del Pezzo surfaces and Mori fiber spaces in positive characteristic, Trans. Am. Math. Soc., Volume 373 (2020) no. 3, pp. 1775-1843 | DOI | MR | Zbl

[GM10] Gessel, Ira M.; Monsky, Paul The limit as p of the Hilbert–Kunz multiplicity of x i d i (2010) | arXiv | Zbl

[Har70] Hartshorne, Robin Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, 156, Springer, 1970 (notes written in collaboration with C. Musili) | DOI | MR | Zbl

[Har77] Hartshorne, Robin Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977 | MR | Zbl | DOI

[Har15] Hara, Nobuo Looking out for Frobenius summands on a blown-up surface of 2 , Ill. J. Math., Volume 59 (2015) no. 1, pp. 115-142 | MR | Zbl

[Hoc10] Hochster, Melvin MATH 615 Lecture Notes, Winter, 2010, 2010 (https://dept.math.lsa.umich.edu/~hochster/615W10/615.pdf)

[Kap86] Kapranov, M. M. The derived category of coherent sheaves on a quadric, Funkts. Anal. Prilozh., Volume 20 (1986) no. 2, p. 67 | DOI | MR | Zbl

[Kaw21] Kawakami, Tatsuro On Kawamata–Viehweg type vanishing for three dimensional Mori fiber spaces in positive characteristic, Trans. Am. Math. Soc., Volume 374 (2021) no. 8, pp. 5697-5717 | DOI | MR | Zbl

[KM08] Kollár, János; Mori, Shigefumi Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, 2008 (with the collaboration of C. H. Clemens and A. Corti, translated from the 1998 Japanese original) | MR | Zbl

[Kol91] Kollár, János Extremal rays on smooth threefolds, Ann. Sci. Éc. Norm. Supér. (4), Volume 24 (1991) no. 3, pp. 339-361 | DOI | MR | Zbl | Numdam

[Kol96] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 32, Springer, 1996 | DOI | MR | Zbl

[Kun69] Kunz, Ernst Characterizations of regular local rings for characteristic p, Am. J. Math., Volume 91 (1969), pp. 772-784 | DOI | MR | Zbl

[Lan08] Langer, Adrian D-affinity and Frobenius morphism on quadrics, Int. Math. Res. Not., Volume 2008 (2008) no. 1, rnm145, 26 pages erratum ibid. 2010, no. 10, 1966-1972 (2010) | DOI | MR | Zbl

[Laz04a] Lazarsfeld, Robert Positivity in algebraic geometry. I Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 48, Springer, 2004 | Zbl | MR

[Laz04b] Lazarsfeld, Robert Positivity in algebraic geometry. II Positivity for vector bundles, and multiplier ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 49, Springer, 2004 | MR | Zbl

[Mab78] Mabuchi, Toshiki C 3 -actions and algebraic threefolds with ample tangent bundle, Nagoya Math. J., Volume 69 (1978), pp. 33-64 | DOI | MR | Zbl

[Mas18] Masaharu, Kaneda On the Frobenius direct image of the structure sheaf of a homogeneous projective variety, J. Algebra, Volume 512 (2018), pp. 160-188 | DOI | MR | Zbl

[Mat89] Matsumura, Hideyuki Commutative ring theory, Cambridge Studies in Advanced Mathematics, 8, Cambridge University Press, 1989 (translated from the Japanese by M. Reid) | Zbl | MR

[Meg98] Megyesi, Gábor Fano threefolds in positive characteristic, J. Algebr. Geom., Volume 7 (1998) no. 2, pp. 207-218 | MR | Zbl

[Mil80] Milne, James S. Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, 1980 | MR | Zbl

[MM81] Mori, Shigefumi; Mukai, Shigeru Classification of Fano 3-folds with B 2 2, Manuscr. Math., Volume 36 (1981) no. 2, pp. 147-162 erratum ibid. 110, p. 407 (2003) | DOI | MR | Zbl

[MM83] Mori, Shigefumi; Mukai, Shigeru On Fano 3-folds with B 2 2, Algebraic varieties and analytic varieties (Tokyo, 1981) (Advanced Studies in Pure Mathematics), Volume 1, North-Holland, 1983, pp. 101-129 | DOI | MR | Zbl

[MM86] Mori, Shigefumi; Mukai, Shigeru Classification of Fano 3-folds with B 2 2. I, Algebraic and topological theories (Kinosaki, 1984), Kinokuniya Company Ltd., 1986, pp. 496-545 | MR | Zbl

[Mor79] Mori, Shigefumi Projective manifolds with ample tangent bundles, Ann. Math. (2), Volume 110 (1979) no. 3, pp. 593-606 | DOI | MR | Zbl

[Mor82] Mori, Shigefumi Threefolds whose canonical bundles are not numerically effective, Ann. Math. (2), Volume 116 (1982) no. 1, pp. 133-176 | DOI | MR | Zbl

[MS78] Mori, Shigefumi; Sumihiro, Hideyasu On Hartshorne’s conjecture, J. Math. Kyoto Univ., Volume 18 (1978) no. 3, pp. 523-533 | DOI | MR | Zbl

[MS03] Mori, Shigefumi; Saito, Natsuo Fano threefolds with wild conic bundle structures, Proc. Japan Acad., Ser. A, Volume 79 (2003) no. 6, pp. 111-114 | DOI | MR | Zbl

[Mur18] Murayama, Takumi Frobenius–Seshadri constants and characterizations of projective space, Math. Res. Lett., Volume 25 (2018) no. 3, pp. 905-936 | DOI | MR | Zbl

[Mur19] Murayama, Takumi Seshadri Constants and Fujita’s Conjecture via Positive Characteristic Methods, Ph. D. Thesis, University of Michigan, Ann Harbor, USA (2019) (https://www.proquest.com/docview/2273364341) | MR

[Ott88] Ottaviani, Giorgio Spinor bundles on quadrics, Trans. Am. Math. Soc., Volume 307 (1988) no. 1, pp. 301-316 | DOI | MR | Zbl

[PSZ18] Patakfalvi, Zsolt; Schwede, Karl; Zhang, Wenliang F-singularities in families, Algebr. Geom., Volume 5 (2018) no. 3, pp. 264-327 | DOI | MR | Zbl

[PW22] Patakfalvi, Zsolt; Waldron, Joe Singularities of general fibers and the LMMP, Am. J. Math., Volume 144 (2022) no. 2, pp. 505-540 | DOI | MR | Zbl

[RŠVdB19] Raedschelders, Theo; Špenko, Špela; Van den Bergh, Michel The Frobenius morphism in invariant theory, Adv. Math., Volume 348 (2019), pp. 183-254 | MR | DOI | Zbl

[Sai03] Saito, Natsuo Fano threefolds with Picard number 2 in positive characteristic, Kodai Math. J., Volume 26 (2003) no. 2, pp. 147-166 | DOI | MR | Zbl

[Sam14] Samokhin, Alexander The Frobenius morphism on flag varieties, I (2014) | arXiv

[Sam17] Samokhin, Alexander The Frobenius morphism on flag varieties, II (2017) | arXiv

[SB97] Shepherd-Barron, Nicholas I. Fano threefolds in positive characteristic, Compos. Math., Volume 105 (1997) no. 3, pp. 237-265 | DOI | MR | Zbl

[Sin05] Singh, Anurag K. The F-signature of an affine semigroup ring, J. Pure Appl. Algebra, Volume 196 (2005) no. 2-3, pp. 313-321 | DOI | MR | Zbl

[ST16] Sannai, Akiyoshi; Tanaka, Hiromu A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf, Math. Ann., Volume 366 (2016) no. 3-4, pp. 1067-1087 | DOI | MR | Zbl

[Tak89] Takeuchi, Kiyohiko Some birational maps of Fano 3-folds, Compos. Math., Volume 71 (1989) no. 3, pp. 265-283 | MR | Zbl

[The21] The Stacks Project Authors Stacks Project, 2021 (http://stacks.math.columbia.edu)

[Tho00] Thomsen, Jesper F. Frobenius direct images of line bundles on toric varieties, J. Algebra, Volume 226 (2000) no. 2, pp. 865-874 | DOI | MR | Zbl

[Tri23] Trivedi, Vijaylaxmi The Hilbert–Kunz density functions of quadric hypersurfaces, Adv. Math., Volume 430 (2023), 109207, 63 pages | DOI | MR | Zbl

[Tuc12] Tucker, Kevin F-signature exists, Invent. Math., Volume 190 (2012) no. 3, pp. 743-765 | DOI | MR | Zbl

[Tyc88] Tyc, Andrzej Differential basis, p-basis, and smoothness in characteristic p>0, Proc. Am. Math. Soc., Volume 103 (1988) no. 2, pp. 389-394 | DOI | MR | Zbl

[Wiś91] Wiśniewski, Jarosław A. On contractions of extremal rays of Fano manifolds, J. Reine Angew. Math., Volume 417 (1991), pp. 141-157 | DOI | MR | Zbl