Article

# A construction of Horikawa surface via Q-Gorenstein smoothings

Mathematische Zeitschrift (Impact Factor: 0.88). 09/2007; DOI: 10.1007/s00209-009-0608-6

Source: arXiv

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**ABSTRACT:**We construct a new family of simply connected minimal complex surfaces of general type with = 1, = 0, and = 3, 4, 5, 6, 8 using a -Gorenstein smoothing theory.Journal of the Korean Mathematical Society 01/2013; 50(3). · 0.32 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**We consider the deformation spaces of some singular product-quotient surfaces $X=(C_1 \times C_2)/G$, where the curves $C_i$ have genus 3 and the group $G$ is isomorphic to $\mathbb{Z}_4$. As a by-product, we give a new construction of Todorov surfaces with $p_g=1$, $q=0$ and $2\le K^2\le 8$ by using $\mathbb{Q}$-Gorenstein smoothings.01/2012; - [Show abstract] [Hide abstract]

**ABSTRACT:**We construct a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=2$ which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with $p_g=0$ and $K^2=1$. In order to construct the example, we combine a double covering and $\mathbb{Q}$-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Koll\'ar and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced $\mathbb{Z}/2\mathbb{Z}$-action.Mathematische Annalen 08/2011; 357(1). · 1.38 Impact Factor

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