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

Bruno De Oliveira, Christopher Langdon

Research output: Contribution to journalArticlepeer-review


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
Issue number2
StatePublished - Jun 15 2019


  • Low codimension
  • Quadrics
  • Symmetric differentials
  • Trisecant variety

ASJC Scopus subject areas

  • Mathematics(all)


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