Local differential geometry and generic projections of threefolds (Q1813982)

The purpose of this note is to prove a result concerning the 4-secant lines of a nondegenerate irreducible, say smooth, threefold \(X \supset \mathbb{P}^ r\), \(r \geq 9\); namely we prove essentially that all these lines together fill up at most a fourfold. Precise statement: Theorem 1. Let \(X\) be an irreducible nondegenerate three-dimensional subvariety of \(\mathbb{P}^ r\), \(r \geq 9\), whose tangent variety is six- dimensional, and let \(\{L_ y:y \in Y\}\) be a family of lines in \(\mathbb{P}^ r\) with the property that for any general \(y \in Y\), the part of the scheme-theoretic intersection \(L_ y\cap X\) supported at smooth points of \(X\) has length at least 4. Then we have \(\dim(\bigcup_{y\in Y}L_ y)\leq 4\). It seems likely that the theorem is true for \(r=7,8\) as well, but the proof does not yield this. -- It is reasonable to except that the analogue of the theorem is true for (nondegenerate) \(n\)-folds \(X\) in \(\mathbb{P}^ r\), \(r \geq 2n+1\): namely that the \((n+1)\)-secant lines of \(X\) fill up at most an \((n+1)\)-fold. Corollary 3. Let \(X\) be a smooth nondegenerate irreducible threefold of degree \(d\) in \(\mathbb{P}^ r\), \(r\geq 9\). Then \(X\) is \((d-r+4)\)-regular, i.e. the ideal sheaf \(I=I_{X/\mathbb{P}^ r}\) satisfies \(H^ i(\mathbb{P}^ r,I(d- r+4-i))=0\) for \(i<0\).