Twisted symmetric differentials and the quadric algebra of subvarieties of P N of low codimension

Bruno De Oliveira, Christopher Langdon

Research output: Contribution to journalArticle

Abstract

A smooth subvariety X⊂ P N has for each α∈ Q an associated algebra Q(X,α)=⨁mα∈ZH0(X,Sm[ΩX1(α)]). The algebra Q(X, 0) is the the intrinsic algebra of symmetric differentials and Q(X, 1) is called the algebra of twisted symmetric differentials. We show that when X is a complete intersection of dimension [InlineEquation not available: see fulltext.], then the algebra of twisted symmetric differentials is the quadric algebra of X, i.e. [InlineEquation not available: see fulltext.]. The same isomorphism is shown without the complete intersection assumption if X is of codimension two and dim X⩾ 3. We establish an identification of the twisted symmetric m-differentials on X with the tangentially homogeneous polynomials relative to X of degree m. The lack of the hypothesis of X being a complete intersection is dealt with results on the properties of the vanishing locus of tangentially homogeneous polynomials and algebraic geometric properties of the tangent-secant variety of X.

Original languageEnglish (US)
Pages (from-to)454-475
Number of pages22
JournalEuropean Journal of Mathematics
Volume5
Issue number2
DOIs
StatePublished - Jun 15 2019

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Quadric
Codimension
Complete Intersection
Q-algebra
Algebra
Homogeneous Polynomials
Secant Varieties
Tangent line
Locus
Isomorphism

Keywords

  • Low codimension
  • Quadrics
  • Symmetric differentials
  • Trisecant variety

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Twisted symmetric differentials and the quadric algebra of subvarieties of P N of low codimension . / De Oliveira, Bruno; Langdon, Christopher.

In: European Journal of Mathematics, Vol. 5, No. 2, 15.06.2019, p. 454-475.

Research output: Contribution to journalArticle

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