Varieties with minimal secant degree and linear systems of maximal dimension on surfaces

By C. Ciliberto, Francesco Russo. 2004

68 citations recorded by the source corpus.

Abstract In this paper we prove, using a refinement of Terracini's Lemma, a sharp lower bound for the degree of (higher) secant varieties to a given projective variety, which extends the well known lower bound for the degree of a variety in terms of its dimension and codimension in projective space. Moreover we study varieties for which the bound is attained proving some general properties related to tangential projections, e.g. these varieties are rational. In particular we completely classify surfaces (and curves) for which the bound is attained. It turns out that these surfaces enjoy some maximality properties for their embedding dimension in terms of their degree or sectional genus. This is related to classical beautiful results of Castelnuovo and Enriques that we revise here in terms of adjunction theory.

Subfields: Computational Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/28b050df1a4b02a294eb8ce66759c269a80884f5

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In Computational Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Varieties with quadratic entry locus, I (2007, methodology): “odimension δ(X) linear subspace of TxXpassing through xis birational. Theorem 2.1 proves one implication, yielding the rationality of QEL-manifolds and extending [CMR, Corollary 4.2]. One may consult [CR] for other generalizations of the above conjecture to higher secant varieties. It is worth menti”
Decomposition of homogeneous polynomials with low rank (2010, background): “By [6], Lemma 2.1.5, or [3], Proposition 11, there is a smoothable 0-dimensional scheme E ⊂ Xm,d such that deg(E) ≤ sbr(P ) and P ∈ 〈E〉.”
On the Concept of k‐Secant Order of a Variety (2006, methodology): “Note that, if s(X) = (k + 1)n + k = r, the k-secant order of X is nothing but the number of (k + 1)-secant P to X passing through the general point of P, that is, the number of apparent (k + 1)-secant Pk−1 of X , which has been considered and studied in [7].”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, unknown): “, dt) in t sets of two variables; here, the generic partially-symmetricrank is t+ 1 [205]; • S2Cn+1 ⊗ S2Cn+1, i.”