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 special Enriques surfaces, we construct new examples of a minimal surface of general type with pg = 0, π1 = Z/2Z, and K 2 ≤ 4.
We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that:-Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism.-Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called P -resolutions of a quotient surface singularity. We provide an explicit algorithm for identifying a given Milnor fiber as a minimal symplectic filling, that is, as complements Z − E ∞ . For this, we apply some techniques from the minimal model program for 3-folds such as divisorial contractions and flips.We first compactify a given smoothing of (X, 0). We briefly sketch the idea: Let M be the Milnor fiber of a smoothing π : X → ∆ of a quotient surface singularity X. Let Y → X be the P -resolution corresponding to π. According to Behnke-Christophersen [3], there is a special partial resolution, so called, M -resolution Y → X dominating Y (See Definition 6.14), and a Q-Gorenstein smoothing φ : Y → ∆ such that the smoothing φ blows down to π. We have a commutative diagram as described in Figure 1. It is easy to show that a general fiber Y t = φ −1 (t) is isomorphic to X t = π −1 (t). Therefore we haveWe then compactify X and Y to compact complex surfaces, following Lisca [26] and Pinkham [41], so that the two smoothings Y → ∆ and X → ∆ can be extended to the deformations Y → ∆ and X → ∆ of the natural compactifications X and Y of X and Y , where X and Y are obtained, roughly speaking, by pasting a regular neighborhood ν(E ∞ ) of the compactifying divisor E ∞ of X (See Section 7 for the definition of the natural compactification). We again have a commutative diagram, as described in Figure 1.The deformations Y → ∆ and X → ∆ are locally trivial along E ∞ . So the Milnor fiber Y t is given as the complement of the compactifying divisor E ∞ in a general fiber Y t (which is called a compactified Milnor fiber ; See Definition 7.6) of the deformation Y → ∆; See Proposition 7.5. So we haveWe need to recognize ( Y t , E ∞ ) as (Z, E ∞ ) in the lists of Lisca [26] and . For this, we show that Main Theorem 2 (Theorem 9.4). By applying specific divisorial contractions and flips in a controlled and explicit manner to the deformation Y → ∆, we obtain a new deformation W → ∆ such that all of its fibers are smooth.During this minimal model program (in short, MMP ) process, we ...
Abstract. Suppose that C = (C 1 , . . . , Cm) is a configuration of 2-dimensional symplectic submanifolds in a symplectic 4-manifold (X, ω) with connected, negative definite intersection graph Γ C . We show that by replacing an appropriate neighborhood of ∪C i with a smoothing W S of a normal surface singularity (S, 0) with resolution graph Γ C , the resulting 4-manifold admits a symplectic structure. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs.
Abstract. We construct a new family of simply connected minimal complex surfaces of general type with pg = 1, q = 0, and K 2 = 3, 4, 5, 6, 8 using a Q-Gorenstein smoothing theory.
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