Illinois Journal of Mathematics

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

Nobuo Hara

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

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