Fibers of projections and submodules of deformations (Q2918453)

Let \(X \subset \mathbb P^r\) be a smooth projective variety of dimension \(n\) defined over an algebraically closed field of characteristic zero, and let \(\pi: X \rightarrow \mathbb P^{n+c}\) be a general linear projection. Based on low dimensional cases and the foundational work of \textit{J. N. Mather} [Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 233--248 (1973; Zbl 0292.58004)] it seems likely that it is possible to bound the degree of a zero-dimensional subscheme \(Y \subset X\) contained in a \(\pi\)-fiber in terms of the invariants \(n\) and \(c\). So far this philosophy works only in special cases, but in an earlier paper [Compos. Math. 146, No. 2, 435--456 (2010; Zbl 1189.14061)] the authors introduced a natural invariant of the \(\pi\)-fiber that is similar to the degree and can always be bounded by \(\frac{n}{c}+1\). In the paper under review the authors generalize their construction and deduce the existence of bounds on the complexity of the fibers. More precisely for \(Y \subset X\) a zero-dimensional subscheme, let NEWLINE\[NEWLINE T_Y := \Hom_{\mathcal O_Y} (\Omega_{Y/k}, \mathcal O_Y) NEWLINE\]NEWLINE be the tangent sheaf. If \(Y\) is contained as a closed subscheme in a fiber of a general linear projection \(\pi: X \rightarrow \mathbb P^{n+c}\), then we have NEWLINE\[NEWLINE \deg Y + \frac{1}{c} \deg T_Y \leq \frac{n}{c}+1. NEWLINE\]NEWLINE Mather's theorem is a corollary of this statement, in particular one obtains that the number of distinct points in every fiber is bounded by \(\frac{n}{c}+1\). We refer to the very interesting introduction for numerous examples and the relation of this result with some difficult open problems in projective geometry.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14002].