اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA characteristic free approach to secant varieties of triple Segre products
184
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases.
قيم البحث
اقرأ أيضاً
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-Hir
schowitz 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.
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.
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 sec
ant 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.
We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold $PP^2timesPP^2$. We characterize the Ulrich bundles $Vv$ of arbitrary rank on $PP^2timesPP^2$ with $h^1(VvotimesOmegaboxtimesOmega)=0$ or with $h^1(VvotimesOmega(-1)b
oxtimesOmega(-1))=0$ or obtained as pullback from $PP^2$ and we construct more complicated examples.
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,
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
