Grassmannians of secant varieties. (Q2753155)

Let \(X\subset {\mathbb P}^r\) be a complex integral non-degenerate variety of dimension \(n\). For \(k\leq r\), a general \((k+1)\)-uple of points in \(X\) spans a \(k\)-plane. Therefore we have the rational map: \(\Phi: X^{k+1} \to G(k,r)\) to the Grassmannian of \(k\)-planes in \({\mathbb P}^r\). By \(G_k(X)\) denote the closure of the image of \(\Phi\) and by \(G_{h,k}(X)\), \(h<k\), the closure of the subset NEWLINE\[NEWLINE \{(L,H) \mid L\in G(h,r), \;H\in G_k(X),\;L\subset H\;\}\subset G(h,r)\times G(k,r). NEWLINE\]NEWLINE \noindent Note that \(G_{0,k}\) is the secant variety \(S_k(X)\), which is the closure of the union of all \(k\)-planes, \((k+1)\)-secant to \(X\). The varieties \(G_{h,k}(X)\) are called Grassmannians of secant varieties. The present paper is devoted to a systematic study of these objects.NEWLINENEWLINEThe expected dimension of \(G_{h,k}(X)\) is equal to NEWLINE\[NEWLINE\min\{(k-h)(h+1)+n(k+1), (r-h)(h+1)\}.NEWLINE\]NEWLINE In the first section it is shown that if this value is not attained, then the dimension of some secant variety is also less than the expected value.NEWLINENEWLINEIn section 2 the case of irreducible surfaces is considered. It is proved that surfaces \(S\subset {\mathbb P}^r\), \(r\geq 5\) with \(\dim G_{1,2}(S)<8\) are either cones or rational normal surfaces of (minimal) degree \(4\) in \({\mathbb P}^5\), but not Veronese surfaces.