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Burniat surfaces. II: Secondary Burniat surfaces form three connected components of the moduli space. (English) Zbl 1219.14051

Invent. Math. 180, No. 3, 559-588 (2010); erratum ibid. 197, No. 1, 237-240 (2014).
A Burniat surface arises as a minimal desingularization of a \(\mathbb{Z}_{2} \times \mathbb{Z}_{2} \)-cover of the projective plane branched in a union of nine lines \(A_{i},B_{i},C_{i},\; i=0,1,2\), where \(A_{0},\;B_{0},\;C_{0}\) form a triangle, \(A_{1},\;A_{2}\) pass through the vertex \(\widehat{A_{0}C_{0}}\), \(B_{1},\;B_{2}\) pass through the vertex \(\widehat{A_{0}B_{0}}\), and \(C_{1},\;C_{2}\) pass through the vertex \(\widehat{B_{0}C_{0}}\). The divisor \(D=\sum A_{i}+\sum B_{i} +\sum C_{i}\) has three \(4\)-fold points in the vertices and \(m\) (\(0 \leq m \leq 4\)) triple points given by extra triads of lines which meet. These configurations of nine lines give rise to surfaces of general type with \(p_{g}=0\) and \(K^{2}=6-m\) [P. Burniat, Ann. Mat. Pura Appl., IV. Ser. 71, 1–24 (1966; Zbl 0144.20203); C. A. M. Peters, Nagoya Math. J. 66, 109–119 (1977; Zbl 0329.14019)].
In [“Burniat surfaces. I: Fundamental groups and moduli of primary Burniat surfaces”, in: C. Faber (ed.) et al., Classification of algebraic varieties. Zürich: European Mathematical Society (EMS). Series of Congress Reports, 49–76 (2011; Zbl 1264.14052)], the first of a series of articles on Burniat surfaces, the authors call a Burniat surface primary if \(K^{2}=6\), secondary if \(K^{2}=5\) or \(4\), tertiary if \(K^{2}=3\) and quaternary if \(K^{2}=2\). In 1994, M. Inoue [Tokyo J. Math. 17 No. 2, 295–319 (1994; Zbl 0836.14020)] constructed examples of surfaces with the same invariants as Burniat surfaces as \(\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times\mathbb{Z}_{2}\)-quotient of an \(\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \mathbb{Z}_{2}\)-invariant divisor of multidegree \((2,2,2)\) in a product of three elliptic curves. In the above cited paper, Bauer and Catanese proved that Burniat surfaces are Inoue surfaces, calculated their fundamental groups and gave an alternative proof of the following result due to M. Mendes Lopes and R. Pardini [Topology 40, No. 5, 977–991 (2001; Zbl 1072.14522)]: The subset of the Gieseker moduli space corresponding to primary Burniat surfaces is an irreducible connected component, normal, rational and of dimension 4. Moreover, they proved that any surface homotopically equivalent to a primary Burniat surface is a primary Burniat surface.
After this explicit description of the moduli space of Burniat surfaces with \(K^2=6\), the authors continue their work in the paper under review, considering secondary Burniat surfaces, i.e. Burniat surfaces with \(K^2=4\) or \(5\). They prove that Burniat surfaces with \(K^2=5\) form an irreducible connected component, normal, rational of dimension 3 of the moduli space \(\mathfrak M^{\min}_{1,5}\) of minimal surfaces of general type with \(\chi=1,\; K^{2}=5\), whereas for surfaces with \(K^{2}=4\), there are two types of line configurations: either there are three collinear points of multiplicity at least 3 for the plane curve formed by the 9 lines and \(K_S\) is not ample (nodal type), or \(K_S\) is ample. For the non-nodal case, they prove that the subset of the moduli space of minimal surfaces of general type \(\mathfrak M^{\min}_{1, 4}\) corresponding to Burniat surfaces with \(K^{2} = 4\) is an irreducible connected component, normal, rational of dimension 2. Moreover, the base of the Kuranishi family of such surfaces \(S\) is smooth. In the nodal case, the subset of the Gieseker moduli space \(\mathfrak M^{\text{can}}_{1, 4}\) of canonical surfaces of general type \(X\) corresponding to Burniat surfaces \(S\) with \(K_S^2 = 4\) is an irreducible connected component of dimension 2, rational and everywhere non-reduced. The corresponding subset of the moduli space \(\mathfrak M^{\min}_{1, 4}\) of minimal surfaces \(S\) of general type is also everywhere non-reduced; the nilpotence order is higher for \(\mathfrak M^{\min}_{1, 4}\). This is another confirmation of R. Vakil’s [Invent. Math. 164, No. 3, 569–590 (2006; Zbl 1095.14006)].
Actually, in the third one of this series of deep and interesting papers [“Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces”, arXiv:1012.3770], the authors correct the above statement describing the whole connected component containing the Burniat surfaces of nodal type with \(K^2=4\). Precisely, they show that the so-called extended Burniat surfaces with \(K^2= 4\), together with the nodal Burniat surfaces with \(K^2= 4\), form an irreducible connected component, normal, unirational of dimension 3 of the Gieseker moduli space \(\mathfrak M^{\text{can}}_{1,4}\). Finally, using results of H. Inose and M. Mizukami [Math. Ann. 244, 205–217 (1979; Zbl 0444.14006)], they prove that Bloch’s conjecture \(A_0(S) = \mathbb{Z}\) for surfaces of general type with \(p_{g}=0\) holds for the family of Burniat surfaces considered in the paper.

MSC:

14J29 Surfaces of general type
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14J10 Families, moduli, classification: algebraic theory
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References:

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