A note on very ample Terracini loci

In this short note we show that, for any ample embedding of a variety of dimension at least two in a projective space, all high enough degree Veronese re-embeddings have non-empty Terracini loci.


Introduction
Terracini loci were introduced by the first author and Chiantini in [2].Their emptiness implies non-defectivity of secant varieties due to the celebrated Terracini's lemma, whereas the converse is not true: there exist non-empty Terracini loci even in the presence of nondefective secants.This triggered the interest for this geometric notion, leading to the results in the aforementioned article.The Terracini locus has been the subject of recent investigations [3,4], especially for Segre and Veronese varieties, that are crucial in the context of tensors.We start off by defining set-theoretically these loci.
Definition.Let X ⊂ P N be a non-degenerate projective variety of dimension n ≥ 1 over an algebraically closed field K. Let S ⊂ X reg be a finite subset of smooth points of X whose cardinality is k.Let (2S, X) be the union of the corresponding 2-fat points (2p, X) supported at the points p ∈ S. Then S is in the kth Terracini locus T k (X) if and only if h 0 (I (2S,X) (1)) > 0 and h 1 (I (2S,X) (1)) > 0. Equivalently, S is in T k (X) whenever the n-dimensional tangent spaces T p X, for p ∈ S, are linearly dependent and their projective linear span is not the ambient space P N .
A consequence of a deep result of Alexander and Hirschowitz [1, Theorem 1.1 and Corollary 1.2] (where in their notation one chooses m = 2) states that for any projective variety X there exists a very ample embedding such that all the secant varieties of X under this embedding are non-defective.The aim of this note is to point out that, even in this very ample regime, the emptiness of the corresponding Terracini locus does not generally hold.Thus we answer in the negative the question whether a statement similar to the one by Alexander and Hirschowitz works for Terracini loci.

Very ample regime
Let K be an algebraically closed field and let X be a projective variety of dimension n over K.We say that an embeddeding X ⊂ P r of X is not secant defective if for each positive integer k the k-secant variety of X has dimension min{r, k(n + 1) − 1}.For a very ample line bundle L on X, let ν L : X − → |L| ∨ denote the associated embedding.The kth secant variety and the kth Terracini locus of ν L (X) are denoted σ k (ν L (X)) and T k (ν L (X)), respectively.We say that ν L (X) is secant non-defective if σ k (ν L (X)) is non-defective for every k ≥ 1.
Theorem 1.Let n ≥ 2 and X be as above.Let F, L ∈ Pic(X), where L is an ample line bundle.Then there exists an integer m 0 (depending only on X, F , L) such that for all m ≥ m 0 the line bundle F + mL is very ample, ν F +mL (X) is secant non-defective, and there Here the cokernel sheaf is either zero or supported on a zero-dimensional scheme.Taking the long exact sequence in cohomology, we then find a surjective map in cohomology ).The zero-dimensional scheme (2S, X) ∩ Y is a closed subscheme of (2S, X) and so we likewise have a surjection Therefore h 1 (I (2S,X) (1)) > 0 too.So any collection of k smooth points of Y ∩ X reg is in the kth Terracini locus of ν F +mL (X).
Remark 2. Let X ⊂ P N be a projective variety with dim X = n ≥ 2 and consider ν d (X).
For any integer k > 0, the set S k ν d (X reg ) of all subsets of ν d (X reg ) with cardinality k is a variety of dimension kn.For d ≫ 0, the families of S ∈ T k (ν d (X)) we found in the proof of Theorem 1 on a fixed curve Y have codimension k in S k ν d (X reg ).Varying Y , we do not decrease significantly the codimension of T k (ν d (X)) in S k ν d (X reg ): the magnitude of this is O(k).We do not have examples for which, when k is increasing with d, T k (ν d (X)) has codimension 1 in S k ν d (X reg ), which is the least codimension allowed in view of the secant non-defectivity result in [1].The case of curves with positive arithmetic genus is treated in the following proposition.
Here different behaviours appear according to the parity of the degree.