61 research outputs found
The Rough Veronese variety
We study signature tensors of paths from a geometric viewpoint. The
signatures of a given class of paths parametrize an algebraic variety inside
the space of tensors, and these signature varieties provide both new tools to
investigate paths and new challenging questions about their behavior. This
paper focuses on signatures of rough paths. Their signature variety shows
surprising analogies with the Veronese variety, and our aim is to prove that
this so-called Rough Veronese is toric. The same holds for the universal
variety. Answering a question of Amendola, Friz and Sturmfels, we show that the
ideal of the universal variety does not need to be generated by quadrics
Collisions of fat points and applications to interpolation theory
We address the problem to determine the limit of the collision of fat points
in $\mathbb{P}^n. We give a description of the limit scheme in many cases, in
particular in low dimension and multiplicities. The problem turns out to be
closely related with interpolation theory, and as an application we exploit
collisions to prove some new cases of Laface-Ugaglia Conjecture.Comment: 19 page
Toric geometry of path signature varieties
In stochastic analysis, a standard method to study a path is to work with its
signature. This is a sequence of tensors of different order that encode
information of the path in a compact form. When the path varies, such
signatures parametrize an algebraic variety in the tensor space. The study of
these signature varieties builds a bridge between algebraic geometry and
stochastics, and allows a fruitful exchange of techniques, ideas, conjectures
and solutions. In this paper we study the signature varieties of two very
different classes of paths. The class of rough paths is a natural extension of
the class of piecewise smooth paths. It plays a central role in stochastics,
and its signature variety is toric. The class of axis-parallel paths has a
peculiar combinatoric flavour, and we prove that it is toric in many cases.Comment: Code for the computations is available at
https://sites.google.com/view/l-colmenarejo/publications/cod
Generic identifiability of pairs of ternary forms
We prove that two general ternary forms are simultaneously identifiable only
in the classical cases of two quadratic and a cubic and a quadratic form. We
translate the problem into the study of a certain linear system on a projective
bundle on the plane, and we apply techniques from projective and birational
geometry to prove that the associated map is not birational
Secant non-defectivity via collisions of fat points
Secant defectivity of projective varieties is classically approached via
dimensions of linear systems with multiple base points in general position. The
latter can be studied via degenerations. We exploit a technique that allows
some of the base points to collapse together. We deduce a general result which
we apply to prove a conjecture by Abo and Brambilla: for and , the Segre-Veronese embedding of in
bidegree is non-defective.Comment: 36 pages, 4 pages, all comments are welcome
Equations of tensor eigenschemes
We study schemes of tensor eigenvectors from an algebraic and geometric
viewpoint. We characterize determinantal defining equations of such
eigenschemes via linear equations in their coefficients, both in the general
and in the symmetric case. We give a geometric necessary condition for a
0-dimensional scheme to be an eigenscheme.Comment: 13 page
Eigenschemes of Ternary Tensors
We study projective schemes arising from eigenvectors of tensors, called
eigenschemes. After some general results, we give a birational description of
the variety parametrizing eigenschemes of general ternary symmetric tensors and
we compute its dimension. Moreover, we characterize the locus of triples of
homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor.
Our results allow us to implement algorithms to check whether a given set of
points is the eigenscheme of a symmetric tensor, and to reconstruct the tensor.
Finally, we give a geometric characterization of all reduced zero-dimensional
eigenschemes. The techniques we use rely both on classical and modern complex
projective algebraic geometry.Comment: The title has been slightly modified. We added two algorithms,
testing whether a given configuration of points in the plane is the
eigenscheme of some tensor, and reconstructing a tensor from its eigenpoint
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