EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π1 = ℤ/2

Title & Authors
EXTENDING HYPERELLIPTIC K3 SURFACES, AND GODEAUX SURFACES WITH π1 = ℤ/2
Coughlan, Stephen;

Abstract
We construct the extension of a hyperelliptic K3 surface to a Fano 6-fold with extraordinary properties in moduli. This leads us to a family of surfaces of general type with $\small{p_g=1}$, q = 0, $\small{K^2=2}$ and hyperelliptic canonical curve, each of which is a weighted complete inter-section inside a Fano 6-fold. Finally, we use these hyperelliptic surfaces to determine an 8-parameter family of Godeaux surfaces with $\small{{\pi}_1={\mathbb{Z}}/2}$.
Keywords
surfaces of general type;Godeaux surfaces;Fano 6-folds;
Language
English
Cited by
References
1.
R. Barlow, Some new surfaces with $p_g=0$, Duke Math. J. 51 (1984), no. 4, 889-904.

2.
R. Barlow, A simply connected surface of general type with $p_g=0$, Invent. Math. 79 (1985), no. 2, 293-301.

3.
F. Catanese and O. Debarre, Surfaces with $K^2=2$, $p_g=1$, q = 0, J. Reine. Angew. Math. 395 (1989), 1-55.

4.
S. Coughlan, Key varieties for surfaces of general type, University of Warwick PhD thesis, 2009.

5.
S. Coughlan, Extending symmetric determinantal quartic surfaces, Geom. Dedicata 172 (2014), 155-177.

6.
Y. Lee and J. Park, A simply connected surface of general type with $p_g=0$ and $K^2=2$, Invent. Math. 170 (2007), no. 3, 483-505.

7.
S. Papadakis and M. Reid, Kustin-Miller unprojection with complexes, J. Algebraic Geom. 13 (2004), no. 2, 249-268.

8.
M. Reid, Surfaces with pg = 0, $K^2=1$, J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 25 (1978), no. 1, 75-92.

9.
M. Reid, Infinitesimal view of extending a hyperplane section-deformation theory and computer algebra, Algebraic geometry (L'Aquila, 1988), 214-286, Lecture Notes in Math., 1417, Springer, Berlin, 1990.

10.
M. Reid, Graded rings and birational geometry, Proc. of algebraic geometry symposium, 1-72, (Kinosaki, Oct 2000), K. Ohno (Ed.), 2000.

11.
B. Saint-Donat, Projective models of K3 surfaces, Amer. J. Math. 96 (1974), 602-639.

12.
C. Werner, A four-dimensional deformation of a numerical Godeaux surface, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1515-1525.