Numerical Gorenstein elliptic singularities

Okuma, Tomohiro

doi:10.1007/s00209-004-0688-2

Cited by 19 publications

(26 citation statements)

References 29 publications

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“…It follows from Grauert-Riemenschneider vanishing theorem (or Lemma 2.7) and [19,Lemma 2.13] that h 1 (O X (−Z 0 )) = h 1 (O X (−C m )) + m = m. As in the proof of Lemma 4.2, we obtain p g (A) = h 1 (O X (−Z 0 )) + 1 = m + 1.…”

Section: Elliptic Singularitiesmentioning

confidence: 56%

Cohomology of ideals in elliptic surface singularities

Okuma¹

2017

Illinois J. Math.

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Section: Elliptic Singularitiesmentioning

confidence: 56%

Cohomology of ideals in elliptic surface singularities

Okuma¹

2017

Illinois J. Math.

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“…The affine piece V 1 ⊂ C 5 of the partial resolution (see [28,Example 6.3]) of (X ′ , o) is defined by the equations sx = y − 3w 2 , sy = z + w 3 , sz = x 2 + 6wy − 2w 3 . Consider the order of the coordinate functions on the exceptional set E ′ on V 1 .…”

Section: A Singularity With P G ≥ 4 Does Not Existmentioning

confidence: 99%

Analytic singularities supported by a specific integral homology sphere link

Némethi

Okuma

2017

Methods and Applications of Analysis

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Abstract. The main question we target is the following: If one fixes a topological type (of a complex normal surface singularity) then what are the possible analytic types supported by it, and/or, what are the possible values of the geometric genus? We answer the question for a specific (in some sense pathological) topological type, which supports rather different analytic structures. These structures are listed together with some of their key analytic invariants.

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“…• Corollary 6.6 of [16] yields that µ = 12m+1. Duco van Straten has used Singular to prove for m = 1, and to indicate the likelihood for general m, that τ = 12m + 1.…”

Section: Normal Surfaces In Cmentioning

confidence: 99%

Milnor and Tjurina numbers for smoothings of surface singularities

Wahl¹

2015

Alg. Geom.

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For an isolated hypersurface singularity {f = 0}, the Milnor number µ is greater than or equal to the Tjurina number τ (the dimension of the base of the semi-universal deformation), with equality if f is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity (V, 0), the difference µ − τ can be defined whether or not (V, 0) is smoothable; the author has proved it is non-negative, and equal to 0 if and only if (V, 0) is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface singularities. Here, µ − τ must be modified so that it is independent of any smoothing. This expression, involving cohomology of exterior powers of the bundle of logarithmic derivations on the minimal good resolution, is conjecturally non-negative, with equality iff (V, 0) is quasi-homogeneous. We prove the "if" part; identify special cases where the conjecture is particularly interesting; verify it in some non-trivial cases; and prove it for a Q-Gorenstein smoothing when the index one cover is a hypersurface. This conjecture arose regarding the classification of surface singularities with rational homology disk smoothings.

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Numerical Gorenstein elliptic singularities

Cited by 19 publications

References 29 publications

Cohomology of ideals in elliptic surface singularities

Cohomology of ideals in elliptic surface singularities

Analytic singularities supported by a specific integral homology sphere link

Milnor and Tjurina numbers for smoothings of surface singularities

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