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Register a new userOn secant defective varieties, in particular of dimension 4
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In this paper we prove some general results on secant defective varieties. Then we focus on the 4--dimensional case and we give the full classification of secant defective 4--folds. This paper has been inspired by classical work by G. Scorza,
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Let $Xsubset mathbb{P}^r$ be an integral and non-degenerate variety. Let $sigma _{a,b}(X)subseteq mathbb{P}^r$, $(a,b)in mathbb{N}^2$, be the join of $a$ copies of $X$ and $b$ copies of the tangential variety of $X$. Using the classical Alexander-Hirschowitz theorem (case $b=0$) and a recent paper by H. Abo and N. Vannieuwenhoven (case $a=0$) we compute $dim sigma _{a,b}(X)$ in many cases when $X$ is the $d$-Veronese embedding of $mathbb{P}^n$. This is related to certain additive decompositions of homogeneous polynomials. We give a general theorem proving that $dim sigma _{0,b}(X)$ is the expected one when $X=Ytimes mathbb{P}^1$ has a suitable Segre-Veronese style embedding in $mathbb{P}^r$. As a corollary we prove that if $d_ige 3$, $1le i le n$, and $(d_1+1)(d_2+1)ge 38$ the tangential variety of $(mathbb{P}^1)^n$ embedded by $|mathcal{O} _{(mathbb{P} ^1)^n}(d_1,dots ,d_n)|$ is not defective and a similar statement for $mathbb{P}^ntimes mathbb{P}^1$. For an arbitrary $X$ and an ample line bundle $L$ on $X$ we prove the existence of an integer $k_0$ such that for all $tge k_0$ the tangential variety of $X$ with respect to $|L^{otimes t}|$ is not defective.
Motivated by the study of the secant variety of the Segre-Veronese variety we propose a general framework to analyze properties of the secant varieties of toric embeddings of affine spaces defined by simplicial complexes. We prove that every such secant is toric, which gives a way to use combinatorial tools to study singularities. We focus on the Segre-Veronese variety for which we completely classify their secants that give Gorenstein or $mathbb Q$-Gorenstein varieties. We conclude providing the explicit description of the singular locus.
It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.
In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo--Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.
In this paper we will prove a uniformity result for the Iitaka fibration $f:X rightarrow Y$, provided that the generic fiber has a good minimal model and the variation of $f$ is zero or that $kappa(X)=rm{dim}(X)-1$.
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