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On the strong base locus of a projective variety

Edoardo Ballico, Maria Chiara Brambilla, Pierpaola Santarsiero

Abstract

We introduce and study the base locus and the strong base locus of a projective variety X. The base locus of X parametrizes configurations of smooth points of X where the span of the tangent spaces of X at these points intersects X at some additional smooth point. The strong base locus parametrizes configurations of smooth points of X for which the span of the tangent spaces of X at the given configuration contains the entire tangent space at an additional point. These notions originate from the study of base loci of tangential projections, are strictly related to interpolation problems with double points in special position, and provide a natural framework to study tangential contact for nongeneral points. We give first properties and explore connections with Terracini loci and with the concept of identifiability. We focus on tensor-related varieties and characterize the nonemptiness of base loci and strong base loci for Veronese and Segre-Veronese varieties.

On the strong base locus of a projective variety

Abstract

We introduce and study the base locus and the strong base locus of a projective variety X. The base locus of X parametrizes configurations of smooth points of X where the span of the tangent spaces of X at these points intersects X at some additional smooth point. The strong base locus parametrizes configurations of smooth points of X for which the span of the tangent spaces of X at the given configuration contains the entire tangent space at an additional point. These notions originate from the study of base loci of tangential projections, are strictly related to interpolation problems with double points in special position, and provide a natural framework to study tangential contact for nongeneral points. We give first properties and explore connections with Terracini loci and with the concept of identifiability. We focus on tensor-related varieties and characterize the nonemptiness of base loci and strong base loci for Veronese and Segre-Veronese varieties.
Paper Structure (6 sections, 15 theorems, 41 equations, 1 figure)

This paper contains 6 sections, 15 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries and first properties of $\mathcal{B}_r(X)$ and $\mathcal{E}_{r}(X)$
  3. Connection with Terracini loci
  4. Veronese varieties
  5. Segre-Veronese varieties
  6. Connection with identifiability

Key Result

Lemma 2.11

Let $Z\subset X$ be a critical scheme for a couple $(A,q)$. Then $\dim \langle Z\rangle =\deg(Z) -2$, $q\in \langle Z\setminus \{q\}\rangle$ and $\dim \langle Z'\rangle = \deg(Z')-1$ for all $Z'\subsetneq Z$.

Figures (1)

  • Figure 1: Possible configurations of $A\in\mathcal{S}_r(\mathbb{P}^n)$ such that $\nu_d(A)\in \mathcal{B}_r(V^d_n)$ with $r\leq d$ from \ref{['theorem: Bb per veronesi']}.

Theorems & Definitions (51)

  • Definition 1.1
  • Definition 1.2
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Example 2.7
  • Remark 2.8
  • Remark 2.9
  • ...and 41 more