On the boundedness of the set of ample vector bundles with fixed sectional genus (Q1313439)

Let \(X\) be a smooth projective variety of dimension \(n\) and \(E\) a vector bundle of rank \(r\) on \(X\). Let \(K_ X\) be the canonical line bundle on \(X\). The \(c_ 1\)-sectional genus of \(E\) is the integer \(g_ c\) defined by the formula \(2g_ c - 2 = (K_ X + c_ 1(E)) c_ 1 (E)^{n-1}\) [\textit{T. Fujita}, J. Math. Kyoto Univ. 29, No. 1, 1-16 (1989; Zbl 0699.14043)]. Let \(A(n, r, g_ c)\) denote the set of pairs \((X,E)\) as above with \(E\) ample and \(g_ c\) fixed. The author shows that if the base field \(K\) is algebraically closed and (1) of characteristic zero, then the set \(A(n,r,g_ c)\) is bounded (i.e. elements of this set `occur' in a family parametrised by an algebraic scheme), (2) if the base field has positive characteristic, then \(A(2,2,g)\) is bounded. In the special case when \(n=r+1\) and \(E\) is spanned by global sections, the author defines the sectional genus \(g_ s\) of \(E\) as the (arithmetic) genus of the zero locus of a general section of \(E\). Note that the zero locus is a curve, smooth in characteristic zero. Let \(Y(3,2,g_ s)\) be the set of pairs \((X,E)\), \(X\) a smooth projective threefold, \(E\) an ample vector bundle of rank two on \(X\) spanned by global sections, with sectional genus \(g_ s\) fixed and \((X, E)\) not a scroll. The author proves that \(Y(3,2,g_ s)\) is bounded if \(\text{char} K = 0\).