arXiv subjects
Publications and source records attributed to Stefano Canino.
We work on two conjectures on tensor eigenvalue multiplicities. By using the language of algebraic geometry we give stronger, more refined versions of the conjectures and we prove them in many new cases, notably for all $2\times 2\times\dots\times 2$ tensors. We also establish a connection between the rank of a tensor and the multiplicities of the zero eigenvalue.
We develop a general first-order theory of Schur apolarity for the study of secant varieties of flag varieties in arbitrary homogeneous embeddings. Extending the classical apolarity--fat-point correspondence for Veronese varieties, we show that in the Schur setting the algebraic square of the apolar ideal need not coincide with the geometric double-point conditions. We introduce a geometric Schur square whose relevant component is the conormal space, yielding a Schur Dual Terracini Lemma. Our construction recovers classical apolarity in the symmetric case. A slot-by-slot Consistency Theorem realizes these intrinsic conditions as multigraded double points. As an application, we determine the dimensions of all secant varieties of $\operatorname{Fl}(1,2;V_n)$ embedded by $\mathcal{O}(1,1)$: the only defective cases are $σ_2(\operatorname{Fl}(1,2;V_3))$ and $σ_3(\operatorname{Fl}(1,2;V_4))$, both of defect one.
We study the postulation of 0-dimensional schemes given by unions of 2-superfat points in general position in the plane, i.e., the union of local schemes defined by the intersection of two distinct double lines. We prove that such schemes have good postulation, i.e., they have the expected Hilbert function. We also show the good postulation of such schemes when we add a general 3-fat point. Finally, we use these results to answer a peculiar kind of interpolation problem.
We consider 0-dimensional schemes supported at a single point in n-space that are m-symmetric, i.e. that intersect any smooth curve passing through the point with length m, and the ones among them that are maximal with respect to inclusion (called m-superfat points). We study properties of such schemes, in particular for n=2. We give a first application of the simplest such schemes, namely 2-superfat points in the plane, by studying varieties defined by them on Veronese and Segre-Veronese varieties and the (symmetric or partially symmetric) tensors they parameterize.
We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the minimum possible value, using very elementary methods, essentially Gröbner basis. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem.
In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces.
The paper is an introduction to the use of the classical Newton-Puiseux procedure, oriented to an algorithmic description of it. This procedure enables to get polynomial approximations for parameterizations of branches of an algebraic plane curve at a singular point. We look for an approach that can be easily grasped and almost self contained. We illustrate the use of the algorithm, first in a completely worked out example of a curve with a point of multiplicity 6, and secondly in the study of triple points on reduced plane curves.