The author of the present paper and F.-O. Schreyer [“Godeaux surfaces. I”, preprint, arXiv:2009.05357] have been studying numerical Godeaux surfaces. We adapt the first few sentences of [loc. cit.].
The minimal surfaces of general type with the smallest possible numerical invariants are the numerical Godeaux surfaces. They have always been of a particular interest in the classification of algebraic surfaces. Y. Miyaoka [Invent. Math. 34, 99–111 (1976; Zbl 0337.14010)] showed that the torsion group of such surfaces is cyclic of order \(m\le 5\). Whereas numerical Godeaux surfaces for \(m= 3, 4, 5\) are completely described by M. Reid [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75–92 (1978; Zbl 0399.14025)], a complete classification in the other cases is still open. Let \(X\) be a numerical Godeaux surface with canonical divisor \(K_X\), and let \(x_0\), \(x_1\) (respectively, \(y_0,\dots,y_3)\) denote a basis of \(H^0(X,\mathcal O_X(2K_X))\) (respectively, of \(H^0(X,\mathcal O_X(3K_X))\)). The canonical ring \(R(X)\) is a quotient of the graded polynomial ring \(\hat S = \pmb k[x_0, x_1, y_0, \dots, y_3, z_0, \dots , z_3,w_0,w_1,w_2]\), where the \(x\)’s have degree two, the \(y\)’s have degree three, the \(z\)’s have degree four, and the \(w\)’s have degree five. That is, there is a closed embedding \[X_{\text{can}} = \operatorname{Proj}(R(X)) \hookrightarrow \operatorname{Proj}(\hat S) = \mathbb P(2^2, 3^4, 4^4, 5^3).\] Thus, one can consider the canonical model of \(X\) as a subvariety of a weighted projective space of dimension \(12\). Alas, studying this embedding is difficult because there is no structure theorem for Gorenstein ideals of such high codimension. Furthermore, from a computational point of view, codimension \(10\) is not promising for irreducibility or non-singularity tests.
As a consequence, Stenger and Schreyer do not consider \(R(X)\) as an \(\hat S\)-algebra. Instead, they view \(R(X)\) as a finitely generated \(S\)-module, where \(S \subseteq \hat S\) is a subring chosen appropriately. Geometrically, this amounts to studying the image of \(X_{\text{can}}\) under the projection into the smaller projective space \(\operatorname{Proj}(S)\). They take \(S\) to be the graded polynomial ring \(S=\pmb k[x_0, x_1, y_0, \dots, y_3]\). The ring \(R(X)\) is finitely generated as a module over \(S\). The structure result from the paper under review, guarantees that the minimal free resolution, \(F\), of \(R(X)\), as an \(S\)-module, is self-dual, of length three, with a skew-symmetric middle map. There are only finitely many isomorphism classes of the complex \(F/(x_0, x_1)\) possible.
This is the starting point of the paper by Stenger and Schreyer.