The variety of bad zero-schemes (Q456811)

Let \(X\) be a smooth complex projective variety of dimension \(n \geq 2\) and an ample base point free linear system \(|V|\) on \(X\). A \textit{bad zero-scheme} for the pair \((X,V)\) is defined by the condition that the sublinear system of elements of \(|V|\) by it contains only reducible or non--reduced elements. In particular one can fix the length of the zero-scheme, say \(t\), and focus on elements of the Hilbert scheme of zero-schemes of length \(t\) of a particular form, for example reduced. This leads the authors to the definition of the \textit{(reduced) \(t\)-bad locus} of the pair \((X,V)\). A count on dimensions on the incidence correspondence correspondence defining these loci provides a bound on the dimension of the reduced \(b_0\)-bad locus, being \(b_0\) the minimal length for which it is non-empty. The bound is reached only when \(n=2\) (see Thm. 3.4) and a description of these surfaces is provided (in particular a classification if \(|V|\) is very ample). Part of these results are extended in Section 4 to the non-reduced case while in Section 5 a new invariant \(s\) is introduced and studied. A library of examples is provided in Section 6.