Twisted symmetric differentials and the quadric algebra of subvarieties of $\mathbb {P}^N$ of low codimension

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.