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Numerical degeneracy of two families of rational surfaces. (English) Zbl 1046.14029

Summary: Let \(X\subset\mathbb{P}^N\) be a closed irreducible \(n\)-dimensional subvariety. The \(k\)-th higher secant variety of \(X\), denoted \(X^k\), is the Zariski closure of the union (in \(\mathbb{P}^N)\) of the linear spaces spanned by \(k\) points of \(X\). A simple dimension count shows that \(\dim X^k\leq k(n+1)-1\) and that when equality holds, there is a non-empty (Zariski) open subset \(U\subset X^k\) and a positive integer \(\sec_k(X)\), such that for all \(z\in U\), there are exactly \(\sec_k(X)\) \(k\)-secant \((k-1)\)-planes to \(X\) through \(z\). Assume that \(\dim X^k= k(n+1)-1\), so that \(\sec_k(X)\) is defined. For \(X^k\) nonlinear we expect \(\sec_k (X)=1\), otherwise we say that \(X^k\) is numerically degenerate. In this paper, the author considers the embeddings \(X\) of \(\mathbb{P}^2\) and \(\mathbb{P}^1 \times \mathbb{P}^1\) by their respective very ample line bundles and classifies those \(k\) for which \(X^k\) is numerically degenerate. In the classification he proves a result of independent interest, showing that a rational normal scroll \(X\) (of arbitrary dimension) never has \(\sec_k(X)>1\).

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N25 Varieties of low degree
14J26 Rational and ruled surfaces
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