Moduli spaces of sheaves in mixed characteristic
The article investigates vector bundles on polarized projective varieties, their Segre invariants and estimates the dimension of their spaces of global sections in terms of \(\mu_{\max}\) -- the maximal slope of a subsheaf. Let \(X\) be a projective variety with a very ample divisor \(H\). Furthermore, let \(E\) be a torsion free sheaf of rank \(r\) on \(X\). If \(\mu_{\max}(E) \geq 0\), then we have the estimate (Theorem 3.3) \[ h^0(X,E) \leq rH^n \cdot \left( \begin{matrix} \frac{\mu_{\max}(E)}{H^n} +f(r) +n \\ n \end{matrix} \right) \quad \text{ with } \quad f(r)=-1+\sum_{k=1}^r \frac{1}{k}. \] The proof uses the Segre invariant which is introduced in the second section. Even though the proof uses well-known techniques as elementary transformations and estimates on binomial coefficients it required ``careful guessing of the statement of the results that we want to prove since the proof breaks down in one of a few steps if we want to prove a weaker or a stronger result'' -- as the author points out. In section 4 this is applied to the construction of moduli spaces of S-equivalence classes of Gieseker semistable sheaves for projective morphisms over a universally Japanese ring. An appendix recalls a result of \textit{S. Ramanan} and \textit{A. Ramanathan} [Tôhoku Math. J., II. Ser. 36, 269--291 (1984; Zbl 0567.14027)] which implies that up to torsion the tensor product of strongly slope semistable sheaves is strongly slope semistable too.
scientific article; zbMATH DE number 2113314