A new example of an algebraic surface with canonical map of degree 16 (Q2274008)

Let \(X\) be a minimal smooth complex surface of general type and \(\varphi_{|K_X|}: x \dashrightarrow \mathbb{P}^{p_g-1}\) be the canonical map. By a result of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006), Proposition 4.1], we see that if \(\dim\varphi_{|K_X|}=2\), the degree \(d\) of the canonical map \(\varphi_{|K_X|}\) is less than or equal to 36. \textit{G. Xiao} later proved that \(d\le8\) if \(p_g>132\) (see [Math. Ann. 274, 473--483 (1986; Zbl 0571.14019), Theorem 3]). There are only a few examples of surfaces with \(d>8\) known so far, such as: Tan's example [\textit{S. Tan}, Math. Ann. 292, No. 1, 13--30 (1992; Zbl 0724.14026)] with \(d = 9\), \textit{U. Persson}'s example [Double coverings and surfaces of general type. Lect. Notes Math. 687, 168--195 (1978; Zbl 0396.14003)] with \(d = 16\), Rito's examples [\textit{C. Rito}, Proc. Am. Math. Soc. 143, No. 11, 4647--4653 (2015; Zbl 1323.14024); Int. J. Math. 28, No. 6, Article ID 1750041, 10 p. (2017; Zbl 1388.14115); Mich. Math. J. 66, No. 1, 99--105 (2017; Zbl 1395.14031)] with \(d = 12\), \(16\), \(24\) and Gleissner, Pignatelli and Rito's examples [``New surfaces with canonical map of high degree'', Preprint, \url{arXiv:1807.11854}] with \(d = 24\), \(q = 1\), and \(d = 32\), \(q = 0\). In this paper, the author construct a new example of surfaces with \(d = 16\), \(K^2_X = 32\), \(p_g = 4\), and \(q = 1\) via the method of abelian cover. The canonical map in this case is a \(\mathbb{Z}^4_2\)-cove of \(\mathbb{P}^1\times \mathbb{P}^1\).