A construction of Horikawa surface via Q-Gorenstein smoothings

Mathematische Zeitschrift (Impact Factor: 0.69). 09/2007; 267(1-2). DOI: 10.1007/s00209-009-0608-6
Source: arXiv


In this article we prove that Fintushel-Stern's construction of Horikawa surface, which is obtained from an elliptic surface via a rational blow-down surgery in smooth category, can be performed in complex category. The main technique involved is Q-Gorenstein smoothings.

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    • "The family of examples described in this section was first introduced by Fintushel and Stern [8], and later reconsidered by Y. Lee and J. Park [20]. "
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    ABSTRACT: We compute the Yamabe invariant for a class of symplectic 4-manifolds of general type obtained by taking the rational blowdown of Kahler surfaces. In particular, for any point on the half-Noether line we exhibit a simply connected minimal symplectic manifold for which we compute the Yamabe invariant.
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    • "Xt , the dimension of the deformation space of X t is equal to h 1 (X t , T Xt ) = 10 − 2K 2 Xt , and hence there is no nontrivial deformation of X t if K 2 Xt ≥ 5. The case of E(n) with n ≥ 4 were worked out in [17] and [14]. The case of E(3) will be treated in the last section. "
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    ABSTRACT: We present methods to construct interesting surfaces of general type via Q-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to Enriques surfaces admitting a special elliptic fibration, we construct minimal surfaces of general type with p_g=0, and \pi_1=Z/2Z. Comment: Small change (Remark 3.11 and references are added.)
    Mathematische Zeitschrift 08/2010; 272(3-4). DOI:10.1007/s00209-012-0985-0 · 0.69 Impact Factor
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    • "where all vertical maps are double covers. Then, it is shown in [10] that the singular surface X n has a Q-Gorenstein smoothing of two quotient singularities simultaneously by Theorem 3.2, and its smoothing is a Horikawa surface (cf. [4]). "
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    ABSTRACT: We show that there is a complex structure on the symplectic 4-manifold $W_{4, k}$ obtained from the elliptic surface E(4) by rationally blowing down $k$ sections for $2\le k\le 9$. And we interpret it via ${\mathbb Q}$-Gorenstein smoothing. This answers affirmatively to a question raised by R. Gompf. Comment: Minor changes are made. It will apper in Advanced Studies in Pure Mathematics (Proceeding of Algebraic Geometry in East Asia)
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