Frobenius direct images of line bundles on toric varieties
Let \(X\) be a scheme over an algebraically closed field of characteristic \(p>0\). Recall that the absolute Frobenius morphism \(F:X\to X\) is the morphism which is the identity on the underlying point set of \(X\) and the \(p\)-th power map at the sheaf level. The author proves that if \(X\) is a smooth toric variety and \(L\) is a line bundle on \(X\), then the direct image \(F_*L\) of \(L\) via \(F_*\) splits into a direct sum of line bundles. This generalizes on old result of Hartshorne for \(X=\mathbb{P}^n\). Thomsen's proof is constructive and gives an algorithm to compute the decomposition explicitly. \textit{R. Bøgvad} [Proc. Am. Math. Soc. 126, No. 12, 3447-3454 (1998; Zbl 0902.14038)] has further generalized this result of Thomsen by showing that for a \(T\)-linearized vector bundle \(M\) in the toric variety \(X\) (under \(T)\), \(F_*M\) splits into a sum of vector bundles of the same rank as \(M\).
scientific article; zbMATH DE number 1448615