We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any ...characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability condition for Gorenstein subschemes holds. The content of the article is largely expository, although many results are presented in a stronger form than in the literature.
We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric ...tensors in C2⊗Cb⊗Cb. We review the literature from a geometric perspective.
Abstract
In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for ...a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.
Some examples of forms of high rank Buczyński, Jarosław; Teitler, Zach
Collectanea mathematica (Barcelona),
09/2016, Volume:
67, Issue:
3
Journal Article
Peer reviewed
We describe some forms with greater Waring rank than previous examples. In 3 variables we give forms of odd degree with strictly greater rank than the ranks of monomials, the previously highest known ...rank. This narrows the possible range of values of the maximum Waring rank of forms in 3 variables. In 4 variables we give forms of odd degree with strictly greater than generic rank. In degrees
≥
5
these are the first examples showing that there exist forms with Waring rank strictly greater than the generic value.
On the locus of points of high rank Buczyński, Jarosław; Han, Kangjin; Mella, Massimiliano ...
European journal of mathematics,
03/2018, Volume:
4, Issue:
1
Journal Article
Open access
Given a closed subvariety
X
in a projective space, the rank with respect to
X
of a point
p
in this projective space is the least integer
r
such that
p
lies in the linear span of some
r
points of
X
. ...Let
W
k
be the closure of the set of points of rank with respect to
X
equal to
k
. For small values of
k
such loci are called secant varieties. This article studies the loci
W
k
for values of
k
larger than the generic rank. We show they are nested, we bound their dimensions, and we estimate the maximal possible rank with respect to
X
in special cases, including when
X
is a homogeneous space or a curve. The theory is illustrated by numerous examples, including Veronese varieties, the Segre product of dimensions (1, 3, 3), and curves. An intermediate result provides a lower bound on the dimension of any
GL
n
orbit of a homogeneous form.
On the third secant variety Buczyński, Jarosław; Landsberg, J. M.
Journal of algebraic combinatorics,
2014/9, Volume:
40, Issue:
2
Journal Article
Peer reviewed
Open access
We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are ...at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show that the singular locus of the secant varieties
has codimension at least two for
r
=2,3.
Waring decompositions of monomials Buczyńska, Weronika; Buczyński, Jarosław; Teitler, Zach
Journal of algebra,
03/2013, Volume:
378
Journal Article
Peer reviewed
Open access
A Waring decomposition of a polynomial is an expression of the polynomial as a sum of powers of linear forms, where the number of summands is minimal possible. We prove that any Waring decomposition ...of a monomial is obtained from a complete intersection ideal, determine the dimension of the set of Waring decompositions, and give the conditions under which the Waring decomposition is unique up to scaling the variables.
We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re‐embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh ...and Stillman (EKS) for curves. We prove that set‐theoretic equations of small secant varieties to a high‐degree Veronese re‐embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However, this is false for singular varieties, and we give explicit counterexamples to the EKS conjecture for singular curves. The techniques that we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud's question in a more general context about the behaviour of border rank under specialization to a linear subspace and provide an overview of conjectures coming from signal processing and complexity theory in this context.
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