Geometric genera for ample vector bundles with regular sections (Q5926242)

The author studies the geometric genus of ample vector bundles \(E\) over smooth projective varieties \(X\). The genus is defined when \(r= \text{rank }(E)\) is smaller than \(n=\dim(X)\) and \(E\) has a section whose \(0\)-locus \(Z\) has the expected codimension \(r\). In this situation one sets \(p_g(E)=h^{n-r,0}(Z)\). Using adjunction theory, one shows that \(p_g(E)\) is always bigger or equal than \(h^{n-r,0}(X)\), so one would like to classify pairs \((X,E)\) for which the equality holds. The author observes that the equality is implied by, and often it is equivalent to, some vanishing conditions on the Koszul cohomology of \(X\) and \(Z\). Using these conditions, the author provides a classification of pairs attaining the equality, under the assumptions that \(r=n-2\) and \(E\) is decomposable.