Moduli spaces of sheaves in mixed characteristic (Q1884507): Difference between revisions

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.