Illinois Journal of Mathematics

Looking out for Frobenius summands on a blown-up surface of $\mathbb{P}^{2}$

Nobuo Hara

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

For an algebraic variety $X$ in characteristic $p>0$, the push-forward $F^{e}_{*}\mathcal{O}_{X}$ of the structure sheaf by an iterated Frobenius endomorphism $F^{e}$ is closely related to the geometry of $X$. We study the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ into direct summands when $X$ is obtained by blowing up the projective plane $\mathbb{P}^{2}$ at four points in general position. We explicitly describe the decomposition of $F^{e}_{*}\mathcal{O}_{X}$ and show that there appear only finitely many direct summands up to isomorphism, when $e$ runs over all positive integers. We also prove that these summands generate the derived category $D^{b}(X)$. On the other hand, we show that there appear infinitely many distinct indecomposable summands of iterated Frobenius push-forwards on a ten-point blowup of $\mathbb{P}^{2}$.

Article information

Source
Illinois J. Math., Volume 59, Number 1 (2015), 115-142.

Dates
Received: 2 March 2015
Revised: 23 November 2015
First available in Project Euclid: 11 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1455203162

Digital Object Identifier
doi:10.1215/ijm/1455203162

Mathematical Reviews number (MathSciNet)
MR3459631

Zentralblatt MATH identifier
1364.14034

Subjects
Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli [See also 14D20, 14F05, 32Lxx]
Secondary: 14G17: Positive characteristic ground fields 14J26: Rational and ruled surfaces

Citation

Hara, Nobuo. Looking out for Frobenius summands on a blown-up surface of $\mathbb{P}^{2}$. Illinois J. Math. 59 (2015), no. 1, 115--142. doi:10.1215/ijm/1455203162. https://projecteuclid.org/euclid.ijm/1455203162


Export citation

References

  • P. Achinger, Frobenius push-forwards on quadrics, Comm. Algebra 40 (2012), no. 8, 2732–2748.
  • P. Achinger, A characterization of toric varieties in characteristic $p$, available at \arxivurlarXiv:1303.5905.
  • M. F. Atiyah, Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3) 7 (1957), 414–452.
  • N. J. Fine, Binomial coefficients modulo a prime, Amer. Math. Monthly 54 (1947), no. 10, 589–592.
  • N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math. 120 (1998), 981–996.
  • N. Hara, T. Sawada and T. Yasuda, $F$-blowups of normal surface singularities, Algebra Number Theory 7 (2013), 733–763.
  • D. Huybrechts, Fourier–Mukai transforms in algebraic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2006.
  • S. Keel, Basepoint freeness for nef and big line bundles in positive characteristic, Ann. of Math. (2) 149 (1999), 253–286.
  • Y. Kitadai and H. Sumihiro, Canonical filtrations and stability of direct images by Frobenius morphisms, Tohoku Math. J. (2) 60 (2008), 287–301.
  • V. B. Mehta and C. Pauly, Semistability of Frobenius direct images over curves, Bull. Soc. Math. France 135 (2007), 105–117.
  • R. Ohkawa and H. Uehara, Frobenius morphisms and derived categories on two-dimensional toric Deligne–Mumford stacks, Adv. Math. 244 (2013), 241–267.
  • A. Sannai and H. Tanaka, A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf, preprint; available at \arxivurlarXiv:1411.5294.
  • K. Schwede, $F$-adjunction, Algebra Number Theory 3 (2009), 907–950.
  • K. E. Smith, Globally $F$-regular varieties: Applications to vanishing theorems for quotients of Fano varieties, Michigan Math. J. 48 (2000), 553–572.
  • K. E. Smith and M. Van den Bergh, Simplicity of rings of differential operators in prime characteristic, Proc. Lond. Math. Soc. (3) 75 (1997), 32–62.
  • X. Sun, Direct images of bundles under Frobenius morphism, Invent. Math. 173 (2008), 427–447.
  • J. F. Thomsen, Frobenius direct images of line bundles on toric varieties, J. Algebra 226 (2000), 865–874.
  • T. Yasuda, Universal flattening of Frobenius, Amer. J. Math. 134 (2012), 349–378.