Minimal algebraic surfaces of general type with \(c_1^2=3\), \(p_g=1\) and \(q=0\), which have non-trivial 3-torsion divisors (Q1890276)

The torsion group of minimal surfaces \(X\) of general type over \({\mathbb C}\) satisfying \(c_1^2(X)< 2\chi({\mathcal {O}}_X)\) is finite. The bounds for the order of the torsion group are well studied for some numerical invariants and very often the existence of torsion allows the classification or construction of examples of such surfaces. Instances of this can be seen in \textit{Y. Miyaoka} [Invent. Math. 34, 99--111 (1976; Zbl 0337.14010)] and \textit{M. Reid} [J. Fac. Sci., Univ. Tokyo, Sect. I A 25, 75--92 (1978; Zbl 0399.14025)] for the case \(p_g=0\), \(c_1^2(X)=1\), \textit{A. Todorov} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 13, No. 1, 1--21 (1980; Zbl 0478.14030)] for the case \(p_g=1\), \(c_1^2(X)=1\). Also the case \( p_g=3\), \(c_1^2(X)=6\) having non-trivial torsion was studied by \textit{A. Bartalesi} and \textit{F. Catanese} [in: Algebraic varieties of small dimension, Proc. Int. Conf., Turin/Italy 1985, Rend. Semin. Mat., Torino, Fasc. Spec., 91--110 (1986; Zbl 0623.14015)], whilst the case \( p_g=1\), \(c_1^2(X)=2\) was studied by \textit{O. Debarre} and \textit{F. Catanese} [J. Reine Angew. Math. 395, 1--55 (1989; Zbl 0658.14016)] and, using these results, \textit{C. Ciliberto} and the reviewer [Geom. Dedicata 66, No. 3, 313--329 (1997; Zbl 0905.14021)] characterized minimal surfaces \(X\) of general type with \(c_1^2(X)= 2\chi({\mathcal {O}}_X)-2\) and non-trivial torsion. The present paper studies minimal algebraic complex surfaces \(X\) of general type satisfying \(c_1(X)^2 = 3,\) \( p_g(X) =1,\) and having non trivial torsion, focusing in particular the case \({\mathbb Z}_3\subseteq\text{Tors}(X)\). It is shown that for minimal surfaces of general type with \(c_1(X)^2 = 3,\) \( p_g(X) =1,\) \( q(X) = 0\), card \( \text{Tors}(X)\leq 4\). Furthermore the case in which the order of the torsion group is \(3\) (i.e \( \text{Tors}(X)= \mathbb{Z}_3 \)) is described by giving a very explicit formulation of the étale triple cover \(Y\to X\) associated to the torsion. The surface \(Y\) is a complete intersection of type \((3,3)\) in \({\mathbb P}^4\) and its equations are explicitely described. The number of moduli of such surfaces \(X\) with ample canonical class is also shown to be \(14\). These results are obtained by first showing, using Noether's inequality, that for these surfaces \(X\), card \( \text{Tors}(X)\leq 6\). Then the author considers the case when \(\mathbb{Z}_3 \subseteq \text{Tors}(X)\) and considers the étale triple cover \(Y\) of \(X\) associated to \({\mathbb Z}_3\). Using the classification results of \textit{K.Konno} [Math. Ann. 290, No. 1, 77--107 (1991; Zbl 0711.14021)], a subtle analysis is done showing that \(Y\) is a complete intersection of type \((3,3)\) in \({\mathbb P}^4\). Since \(Y\) is simply connected, necessarily \(\text{Tors}(X)= {\mathbb Z}_3\). In particular card \( \text{ Tors}(X)\leq 5\). Finally the case card \( \text{Tors}(X)\leq 5\) is excluded by similar methods, using this time classification results of \textit{E. Horikawa} [Invent. Math. 37, 121--155 (1976; Zbl 0339.14025)].