## 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 language | English (US) |
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Pages (from-to) | 454-475 |

Number of pages | 22 |

Journal | European Journal of Mathematics |

Volume | 5 |

Issue number | 2 |

DOIs | |

State | Published - Jun 15 2019 |

## Keywords

- Low codimension
- Quadrics
- Symmetric differentials
- Trisecant variety

## ASJC Scopus subject areas

- Mathematics(all)