On subcanonical Gorenstein varieties and apolarity (Q2840168)

Let \(X\) be a codimension \(1\) subvariety of dimension greater than \(1\) of a variety of minimal degree \(Y.\) If \(X\) is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then \(X\) is arithmetically Gorenstein and the authors characterize such subvarieties \(X\) of \(Y\), via apolarity, as those whose apolar hypersurfaces are Fermat.NEWLINENEWLINEIn this paper, the authors extensively use the concept of an arithmetically Cohen-Macaulay projective variety (aCM for short), and the concept of an arithmetically Gorenstein variety (aG for short).NEWLINENEWLINEThey extend the results of \textit{M. L. Green} [Duke Math. J. 49, 1087--1113 (1982; Zbl 0607.14005)] with the use of the general Kodaira vanishing theorem. They consider normal projective \(n\)-dimensional aCM varieties \(X \subset \mathbb{P}^N\) with canonical Gorenstein singularities that are regular (i.e. \(h^{1,0}(X) = 0\)) for which \(\omega_X\) is base-point free, the image of the canonical map \(\phi_{\omega_X}\) has maximal dimension and \(h^0(X, \omega_X) \geq n + 2\). They show that, for such varieties, the canonical ring \(R\) of \(X\) is generated in degree \(n\) unless the image of the canonical map is a variety of minimal degree, in which case \(R\) is generated by elements of degree at most \(n + 1\). The above results are also generalized to \(s\)-subcanonical varieties; see Proposition 1.NEWLINENEWLINEWe also give conditions under which a projective variety \(X \subset \mathbb{P}^N\) with canonical Gorenstein singularities is aG. We show (See Theorem 6 and Theorem 8) that this happens if \(X\) is s-subcanonical and \(\ell\)-normal, for all \(\ell\) with \(0 \leq \ell \leq n + s -1\) and, if \(s \geq 0\), satisfies the additional condition that \(h^i(X,O_X(k)) = 0\) for \(1 \leq i \leq n-1\) and \(0 \leq k \leq s\).NEWLINENEWLINEIn this paper, the authors also prove natural generalizations of the celebrated Noether and Enriques-Petri-Babbage theorems in the wider context of s- subcanonically regular varieties (See Theorem 23).NEWLINENEWLINEThe main theorem is the first step to study the geometry of an \(s\)- subcanonically regular variety of dimension \(n\) via the behaviour of the rational map \(\alpha_X : G(m,N) \rightarrow H_{m,s+n+1}\). For a non-trivial example concerning the canonical curve case see \textit{E. Ballico, G. Casnati} and \textit{R. Notari} [J. Algebra 332, No. 1, 229--243 (2011; Zbl 1242.14030)].NEWLINENEWLINEThe assumption that the resolution is crepant establishes an interesting link between the theory of singularities and the theory of apolarity. Moreover, some of the geometry described in this paper could shed some light on some aspects of Artinian Gorenstein Rings, see [\textit{A. Conca, M. E. Rossi} and \textit{G. Valla}, Compos. Math. 129, No. 1, 95--121 (2001; Zbl 1030.13005)].