Regular threefolds of general type with \(p_g =0\) and large bigenus
This paper continues from and improves on a previous publication [Boll. Unione Mat. Ital. (9) 2, No. 3, 607--621 (2009; Zbl 1184.14069)] in which the author constructed threefolds of general type with \(q_1=q_2=p_g=0\) and \(P_2=6\) and with the \(m\)-canonical map birational for \(m \geq 2\). In the paper under review, two new threefolds of general type are constructed with \(q_1=q_2=p_g=0\) and \(P_2=7,8\) and with the \(m\)-canonical map birational for \(m \geq 2\) . The threefolds are defined over the complex numbers. The construction of these threefolds is a contribution to the problem of establishing all the possible values of the bigenus \(P_2\) of threefolds of general type with \(q_1=q_2=p_g=0\). While for the case of (regular) surfaces with \(p_g=0\), we know that \(P_2\) satisfies \(2 \leq P_2 \leq 10\), and that \(P_2\) can have all the values from \(2\) to \(10\), in the case of threefolds and in the light of the paper under review, we know that \(P_2\) can take all the values from \(0\) to \(8\), but we do not know whether or not it admits an upper bound \(< \infty\). Examples of threefolds with \(q_1=q_2=p_g=0\) and \(P_2=0,1,\dots,6\) are due to M. Reid, A.R. Iano-Fletcher, the author of the paper under review, R. Gattazzo, and the reviewer. For all references, cf. the previously quoted review and [\textit{R. Gattazzo}, Int. J. Pure Appl. Math. 66, No. 3, 339--353 (2011; Zbl 1218.14008)]. We note explicitly that, if we do not need the first irregularity to be \(q_1=0\) (as required in order to have an analogy with the case of surfaces), then it is easy to construct examples with geometric genus \(p_g=0\) and bigenus \(P_2\) as large as we like, considering the product of suitable curves and surfaces.
scientific article; zbMATH DE number 6107834
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