Frobenius splitting and derived category of toric varieties
Let \(X\) be a smooth complete toric variety over an algebraically closed field of characteristic \(0\). \textit{A. King} [``Tilting bundles on some rational surfaces'', \url{http://www.maths.bath.ac.uk/~masadk/papers/tilt.pdf}] conjectured that \(X\) has a full strongly exceptional collection of line bundles, that is, that the bounded derived category \(D(X)\) of complexes of coherent sheaves on \(X\) admits a semiorthogonal decomposition by exceptional line bundles with extra orthogonality properties. A counterexample to this conjecture is provided by \textit{L. Hille} and \textit{M. Perling} [Compos. Math. 142, No. 6, 1507--1521 (2006; Zbl 1108.14040)]: for a blow up of three points in the degree 2 Hirzebruch surface, the exceptional collections fail to be strong. The authors notice that the counterexample is not a Fano variety and then they conjecture King's statement to hold for any smooth complete Fano toric variety over an algebraically closed field of characteristic \(0\). In this paper, they prove that King's conjecture holds indeed for such varieties of (almost) maximal Picard number: if \(X\) is \(n\)-dimensional, and \(n\) odd, for \(\rho_X = 2n-1\), if \(X\) is \(n\)-dimensional, and \(n\) even, for \(\rho_X = 2n\) and \(\rho_X = 2n-1\). The main ingredients in the proof are the classification of smooth complete Fano toric varieties with such Picard numbers, and a case-by-case analysis of the splitting of \((\pi_p)_*({\mathcal O}_X)\), where \(\pi_p\) is the Frobenius morphism.
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