The (dimension \(+2\))-secant lemma
The classic and important trisecant lemma states that a non-singular nondegenerate curve \(C\) in \(\mathbb{P}^ r\), \(r \geq 3\), admits at most \(\infty^ 1\) trisecant lines. (A few low-degree curves, such as a rational normal curve or more generally one whose homogeneous ideal is generated by quadrics, have no trisecant lines.) A slightly weaker statement of this theorem is that the generic projection of \(C\) to \(\mathbb{P}^2\) has at most double points; that is, the union of the trisecants is at most two-dimensional. The present paper explores a higher dimensional analog of this theorem. The first statement of the trisecant lemma above does not extend in the ``obvious'' way since it is known that there are examples of \(n\)- dimensional varieties admitting \(\infty^{n+1}\) \(k\)-secant lines, for any \(k\). However, the union of these lines in these examples is still only \((n+1)\)-dimensional. The author thus takes the point of view of the slightly weaker statement above. He proves that if \(\dim X = n\) then the \((n+2)\)-secant lines span at most an \((n+1)\)-fold. In fact, he proves something much more general: Let \(X \subset {\mathbb{P}}^r\) be a (possibly singular and/or reducible) \(n\)-dimensional subvariety, and let \(Y\) be an irreducible variety parameterizing a family \(\{L_y \}\) of \(\ell\)-dimensional linear subspaces of \(\mathbb{P}^r\), \(1 \leq \ell \leq r-n\). Assume that a general \(L_y\) has the property that the part of the scheme-theoretic intersection \(L_y \cap X\) supported at smooth points of \(X\) contains a curvilinear open subscheme of length at least \(n+2\). Then a general \((\ell -1)\)-dimensional linear subspace of \({\mathbb{P}}^ r\) is not contained in any \(L_ y\). The author shows how to arrive from this theorem to the result mentioned above about \((n+2)\)-secant lines, and he also obtains several other corollaries of his theorem.
scientific article; zbMATH DE number 59793