Canonical surfaces in \(\mathbb P^4\) with \(p_g=p_a=5\) and \(K^2=11\)
The author uses his results characterising Gorenstein algebras in codimension two [J. Algebra 288, No. 2, 545--555 (2005; Zbl 1099.13040)] to obtain information about the moduli of general type surfaces with \(p_g=5\), \(K^2=11\) and \(q=0\). The study proceeds via images of the surfaces in \({\mathbb P}^4\) by linear subsystems of the canonical system. Suppose then that \(S\) is a smooth regular surface and \(L\subset H^0({\mathcal O}_S(K_S))\) is a \(5\)-dimensional base-point free linear subspace such that the corresponding morphism \(\pi \colon S\to {\mathbb P}^4\) is birational onto its image~\(Y\). Then \(S\) is a minimal surface of general type and its canonical ring \({\mathcal R}\) is a Gorenstein \({\mathcal A}\)-algebra of codimension~\(2\), where \({\mathcal A}\) is the homogeneous coordinate ring of \({\mathbb P}^4\). One can deduce the shape of a minimal resolution of \({\mathcal R}\) as an \({\mathcal A}\)-algebra. In the case \(p_g=5\), \(K^2=11\), \(q=0\) one can be more precise, calculating the twists involved exactly. In this case there is a converse, which is proved here (Theorem~1.6): an \({\mathcal A}\)-algebra \({\mathcal R}\) with resolution of this shape and satisfying mild depth and singularity conditions arises this way. The author then considers this situation and writes down in detail the condition on the resolution imposed by the assumption that \({\mathcal R}\) is a ring. This yields (rather laboriously) a standard form of the matrices involved and hence a parametrisation of the moduli of the surfaces. One obtains a \(38\)-dimensional unirational open subset of the moduli of surfaces of general type with \(p_g=5\), \(K^2=11\) and \(q=0\), consisting of those whose canonical map is a birational morphism that satisfy a genericity assumption. There remains the need to check that this subset is nonempty, which is done by a computer calculation, sketched here but not given in every detail.
scientific article; zbMATH DE number 5292351