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Terracini Loci: Dimension and Description of Its Components

Author

Listed:
  • Edoardo Ballico

    (Department of Mathematics, University of Trento, 38123 Povo, TN, Italy
    The author is a member of GNSAGA of INdAM (Italy).)

Abstract

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X , we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions.

Suggested Citation

  • Edoardo Ballico, 2023. "Terracini Loci: Dimension and Description of Its Components," Mathematics, MDPI, vol. 11(22), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:22:p:4702-:d:1283792
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