### 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) |
---|---|

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 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

^{N}of low codimension ',

*European Journal of Mathematics*, vol. 5, no. 2, pp. 454-475. https://doi.org/10.1007/s40879-018-0265-6

}

TY - JOUR

T1 - Twisted symmetric differentials and the quadric algebra of subvarieties of P N of low codimension

AU - De Oliveira, Bruno

AU - Langdon, Christopher

PY - 2019/6/15

Y1 - 2019/6/15

N2 - 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.

AB - 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.

KW - Low codimension

KW - Quadrics

KW - Symmetric differentials

KW - Trisecant variety

UR - http://www.scopus.com/inward/record.url?scp=85065195730&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85065195730&partnerID=8YFLogxK

U2 - 10.1007/s40879-018-0265-6

DO - 10.1007/s40879-018-0265-6

M3 - Article

AN - SCOPUS:85065195730

VL - 5

SP - 454

EP - 475

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 2

ER -