Normal two-dimensional hypersurface triple points and the Horikawa type resolution

Ashikaga, Tadashi

doi:10.2748/tmj/1178227335

Cited by 13 publications

(46 citation statements)

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“…Our strategy is similar to the canonical resolution for double and triple covers. So let σ (1) : Y (1) → Y be the blowing-up at a singularity of (π). Then we obtain the following diagram of normalizations of Y (1) .…”

Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

“…P (4) σ (1) σ (2) σ (3) Here each number is the valuation at each component. Then P (2) ∈ Y (1) and P (4) ∈ Y (3) are intersections of divisors with the valuations two and one (mod 4), and P (4) is an infinitely near point of P (2) .…”

Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

“…π (1) φ (1) By the succession of this process, the branch locus is modified to smooth for double and triple covers. However, we see, from the following example, that it is not true for 4-fold covers.…”

Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

“…Let π : X → Y be a 4-fold cover. Then there is a succession of blowingups, σ = σ (1) • · · · • σ (r) …”

Section: Introductionmentioning

confidence: 99%

“…Triple covers are studied by many authors [1,4,7,8,9]. Among them, Tan's canonical resolution [8] is useful for the study of triple covers.…”

Section: Introductionmentioning

confidence: 99%

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On 4-Fold Covers of Algebraic Surfaces

Shirane

2010

Kyushu J. Math.

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Abstract. In 1975 Horikawa introduced a method of resolving singularities of double covers over a smooth surface, called the canonical resolution. Ashikaga gave a similar method for certain triple covers in 1992, and Tan constructed the canonical resolution for any triple covers in 2002. These methods are useful for the global or local study of branched covers of surfaces. In this paper, we consider similar resolution for 4-fold covers over a smooth surface, which is based on Lagrange's method to solve quartic equations. By using this method, we compute the Chern numbers c 2 1 and c 2 of certain 4-fold covers over a smooth projective surface. IntroductionIn this paper, all varieties are defined over the field of complex numbers C. Let X and Y be normal projective varieties, and π : X → Y a finite surjective morphism. We call π (or X) a 4-fold cover of Y if deg π = 4. We denote the rational function fields of X and Y by C(X) and C(Y ), respectively. Under these circumstances,Let G be a finite group. If C(X)/C(Y ) is a Galois extension with Gal(C(X)/C(Y )) ∼ = G, then we simply call π a G-cover. We denote the branch locus of π by (π). Horikawa's canonical resolution [3] has been intensively used to study double covers. Triple covers are studied by many authors [1, 4, 7, 8, 9]. Among them, Tan's canonical resolution [8] is useful for the study of triple covers. However 4-fold covers are not well understood. In [10], Tokunaga studied S 4 -covers based on Lagrange's method of solving quartic equations, where S 4 is the symmetric group of degree four (this method is based on the symmetry of roots; see [6, pp. 266-270] for details). Our approach of 4-fold covers is based on [10].Throughout this paper, we assume that the base space Y is smooth. Let π : X → Y be a 4-fold cover. Then there is an element z of C(X) such that its minimal polynomial overwhere g i (i = 1, 2, 3) are elements of C(Y ). Let K be the Galois closure of C(X)/C(Y ). In this paper, we always assume that Gal( K/C(Y )) ∼ = S 4 (see Remark 4.1 for other cases).By [10], we obtain the diagram of the field extensions and the normalizations of Y as in Figure 1. Hereπ (respectively ϕ) is an S 4 -(respectively S 3 -) cover, ψ 1 is a double cover, ψ 2 is a cyclic triple cover, and ψ 3 is a bidouble cover (i.e. V 4 -cover, where V 4 := {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} ∼ = (Z/2Z) ⊕2 ).

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Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

Section: Resolution Methods For 4-fold Coversmentioning

confidence: 99%

“…Let π : X → Y be a 4-fold cover. Then there is a succession of blowingups, σ = σ (1) • · · · • σ (r) …”

Section: Introductionmentioning

confidence: 99%

“…Triple covers are studied by many authors [1,4,7,8,9]. Among them, Tan's canonical resolution [8] is useful for the study of triple covers.…”

Section: Introductionmentioning

confidence: 99%

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On 4-Fold Covers of Algebraic Surfaces

Shirane

2010

Kyushu J. Math.

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The Inequality 8p_g < μ for Hypersurface Two‐dimensional Isolated Double Points

Tomari

1993

Mathematische Nachrichten

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The Milnor number p and the geometric genus pB of normal 2-dimensional double points are studied by using Zariski's canonical resolution. By using formulas due to E. HORIKAWA and H. LAUFER, we represent p -8pg in terms of the number of blowing-ups along IP' and the number I of "even" components in the resolution process. A key point of our arguments is the fact that if I is small then the resolution process is restricted very much. For rational double points and double points with p , = 1, each classes are characterized by numerical invariants appearing in this resolution process. For the case p , = 1, we can make our inequality sharper and can prove 12. pB -3 < p. This is an another proof of Xu-Yau's inequality for the singularity with p , = 1 in our situation.(l985), 297-354 9-329

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