The paper is devoted to the construction of algebraic varieties using commutative algebraic methods. The varieties of interest are curves, surfaces, \(3\)-folds, and results of Enriques and Fano. The geometric constructions of Enriques, Horikawa and others can often be interpreted in algebraic terms as contructions of rings by generators and relations; for example, the geometric idea of projection corresponds algebraically to elimination of variables.
In particular, if \(R=\sum_{n\geq 0}R_{n}\) is a graded ring, then \(X= \operatorname {Proj} R\) is defined as the quotient \((\operatorname {Spec} R\setminus 0)/\mathbb{C}^{*}\) of the variety \(\operatorname {Spec} R = V(I)\subseteq \mathbb{C}^{n+1}\) by the action of the multiplicative group \(\mathbb{C}^{*}\) induced by the grading. Let \(S\) be a general surface, assume \(K_{S}\) ample, and that \(q=h^{1}(S, O_{S})=0\). Some classes of varieties were treated geometrically by Enriques, Kodaira, Horikawa; algebraic treatment by Ciliberto, Catanese, Reid and others.
If \(R\) is Cohen-Macaulay or Gorenstein, it is easier to construct algebraic varieties. The cone over a projectively embedded abelian surface is a simple example of a non Cohen-Macaulay variety. Horikawa’s study of surfaces with \(p_{g}=4\) and \(K^2 = 6\) divides them into several cases, and solves many problems. But it leaves the existence of degenerations between the following two cases:
\(1)\) the case where \(| K_{X}| \) is a free linear system and defines a \(3\)-to-\(1\) morphism \(\phi_{K_{X}}:X\to Q\), where \(Q\) is the quadric cone \(x_1 x_3=x_2^2\), and
\(2)\) the case where \(| K_{X}| \) has a double point as its base locus on the canonical model, and \(\phi_{{X}}:\widetilde{X}\to Q\), is a \(2\)-to-\(1\) morphism to the quadric cone.
Bauer, Catanese and Pignatelli have recently proved that such a degeneration does occur. Surfaces with \(p_g=1\) and \(K^2=2\) were studied by Catanese and Debarre, following Enriques; an alternative construction is given by Jan Stevens.
For the entire collection see [Zbl 1064.00004].