Abstract
A smooth subvariety \(X\subset \mathbb {P}^N\) has for each \(\alpha \in \mathbb Q\) an associated algebra \(Q(X,\alpha )\!\!=\bigoplus _{m\alpha \in \mathbb Z}H^0(X,S^m[\Omega _X^1(\alpha )])\). 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 , then the algebra of twisted symmetric differentials is the quadric algebra of X, i.e. . The same isomorphism is shown without the complete intersection assumption if X is of codimension two and \(\dim X\geqslant 3\). 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.
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De Oliveira, B., Langdon, C. Twisted symmetric differentials and the quadric algebra of subvarieties of \(\mathbb {P}^N\) of low codimension. European Journal of Mathematics 5, 454–475 (2019). https://doi.org/10.1007/s40879-018-0265-6
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DOI: https://doi.org/10.1007/s40879-018-0265-6