Twisted symmetric differentials and the quadric algebra of subvarieties of \(\mathbb{P}^N\) of low codimension
Let \(X \subset {\mathbb P}^N\) be a smooth complex non-denenerate projective variety and \(\Omega_X^1\) its cotangent bundle. The algebra \(Q(X,1)\) of twisted symmetric differentials is defined as \(\bigoplus_{m \in {\mathbb Z}}H^0(X, S^m[\Omega^1(1)])\), where \(S^m\) is standing for the \(m\)-symmetric power. In the paper under review it is shown (see Theorem 3.1) that if \(X \subset {\mathbb P}^N\) is a complete intersection and \(\dim X>\max\{1,2(N-1)/3\}\) then there exists a graded isomorphim between \(Q(X,1)\) and the so called quadric algebra \(QA(X)\), which is the algebra generated by quadratic polynomials vanishing at \(X\), i.e., \(QA(X)=\mathrm{Sym}(H^0({\mathbb P}^N, {\mathcal I}_X(2)))\). In view of Hartshorne's conjecture on complete intersection one can conjecture (see Conjecture 1.1) that the hypothesis of \(X \subset {\mathbb P}^N\) being a complete intersection can be dropped out. The authors prove this conjecture for codimension 2 and \(\dim X\geq 3\). The algebra \(Q(X,1)\) encodes (first order infinitesimal) information about the embedding \(X \subset {\mathbb P}^N\); in Theorem 2.12 it is shown that in the same range of dimensions \(n >\max\{1,2(N-1)/3\) the algebra \(Q(X,1)\) is in fact graded isomorphic to the subalgebra of \({\mathbb C}[X_0, \dots, X_N]\) generated by polynomials that are tangentially homogeneous relative to \(X\), which means that for all \(x \in X\) the polynomial is vanishing on a cone on the embedded tangent space to \(X\) at \(x\).
scientific article; zbMATH DE number 7073692
scientific article