Ranks of maps of vector bundles (Q6624989)

The author generalizes to vector bundles the techniques introduced for line bundles in: [\textit{F. Liu} et al., Trans. Am. Math. Soc. 374, No. 1, 367--405 (2021; Zbl 1453.14026)]. In section 2 one proves, with some numerical restrictions in Theorem 2.1, that the generic curve \(C\) has a component of expected dimension of the Brill-Noether locus of vector bundles of rank \(r\) degree \(d\) with \(k\) sections for which the Petri map is generically injective. In section 3 assume the characteristic is zero. For \(E\) a vector bundle on \(C\) one has the decomposition \(E\otimes E^*\cong\mathcal{O}\oplus Tr_0E\), where \(Tr_0E\) is the set of traceless endomorphisms. The natural cup-product map for \(E\otimes E^*\) decomposes as a direct sum of two maps. The first map is onto if and only if \(C\) is not hyperelliptic. One proves that the second map is onto for the general curve and general vector bundle.