Abundance theorem for log surfaces
We establish the abundance theorem for log surfaces without assuming $\mathbb{Q}$-factoriality or log canonicity.
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arXiv subjects
Makoto Enokizono
Publications and source records attributed to Makoto Enokizono.
Slope inequality of fibered surfaces, Morsification conjecture and moduli of curves
Using the theory of moduli of curves, we establish various slope inequalities for general fibered surfaces. More precisely, we introduce the notion of functorial divisors on Artin stacks and prove a theorem concerning their effectiveness. Considering the above concept with the Morsification conjecture and the semistable reduction, we obtain several slope (in)equalities, e.g., a generalization of Moriwaki's slope inequality, slope equalities of general fibered surfaces whose fibers satisfy the Morsification conjecture. As applications, we provide a positive answer to Reid's conjecture concerning algebraic Morsification of non-hyperelliptic fibrations of genus 3, a positive partial answer to the question posed by Lu and Tan regarding the Chern invariants of fiber germs, and a partial result concerning lower bounds of the slope of effective divisors on the moduli spaces of stable curves.
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Minimal model program for log canonical pairs on complex analytic spaces
We study the minimal model program for lc pairs on projective morphism between complex analytic spaces. More precisely, we generalize the results by Birkar and the second author to the setup by Fujino.
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On termination of minimal model program for log canonical pairs on complex analytic spaces
We study the termination of minimal model programs for log canonical pairs in the complex analytic setting. By using the termination, we prove a relation between the minimal model theory for projective log canonical pairs and that for log canonical pairs in the complex analytic setting. The minimal model programs for algebraic stacks and analytic stacks are also discussed.
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Uniform bases for ideal arrangements
In this paper we introduce and study uniform bases for the ideal arrangements in all Lie types. Explicit uniform bases are given by Abe-Horiguchi-Masuda-Murai-Sato for types $A,B,C,G$ and we provide them for other types. Combining the explicit uniform bases with the work of Abe-Horiguchi-Masuda-Murai-Sato, we also obtain explicit presentations of the cohomology rings of regular nilpotent Hessenberg varieties in all Lie types.
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Normal stable degenerations of Noether-Horikawa surfaces
We classify all normal stable Horikawa surfaces with only $\mathbb{Q}$-Gorenstein smoothable log canonical singularities. Furthermore, we provide a criterion for their global $\mathbb{Q}$-Gorenstein smoothability and describe the boundary strata of the moduli space of $\mathbb{Q}$-Gorenstein smoothable normal stable Horikawa surfaces.
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Semistable reduction for complex analytic spaces
Semistable reduction theorem for projective morphisms in the category of complex analytic spaces is established.
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An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties
In this paper we construct an additive basis for the cohomology ring of a regular nilpotent Hessenberg variety which is obtained by extending all Poincaré duals of smaller regular nilpotent Hessenberg subvarieties. In particular, all of the Poincaré duals of smaller regular nilpotent Hessenberg subvarieties are linearly independent.
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Vanishing theorems and adjoint linear systems on normal surfaces in positive characteristic
We prove the Kawamata-Viehweg vanishing theorem for a large class of divisors on surfaces in positive characteristic. By using this vanishing theorem, Reider-type theorems and extension theorems of morphisms for normal surfaces are established. As an application of the extension theorems, we characterize non-singular rational points on any plane curve over an arbitrary base field in terms of rational functions on the curve.
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An integral version of Zariski decompositions on normal surfaces
We show that any pseudo-effective divisor on a normal surface decomposes uniquely into its "integral positive" part and "integral negative" part, which is an integral analog of Zariski decompositions. By using this decomposition, we give three applications: a vanishing theorem of divisors on surfaces (a generalization of Kawamata-Viehweg and Miyaoka vanishing theorems), Reider-type theorems of adjoint linear systems on surfaces (including a log version and a relative version of the original one) and extension theorems of morphisms defined on curves on surfaces (generalizations of Serrano and Paoletti's results).
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Durfee-type inequality for hypersurface surface singularities
We prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
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Durfee-type inequality for complete intersection surface singularities
We prove that the signature of the Milnor fiber of smoothings of a $2$-dimensional isolated complete intersection singularity does not exceed the negative number determined by the geometric genus, the embedding dimension and the number of irreducible components of the exceptional set of the minimal resolution, which implies Durfee's weak conjecture and a partial answer to Kerner--Némethi's conjecture.
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Slope equality of plane curve fibrations and its application to Durfee's conjecture
We give a slope equality for fibered surfaces whose general fiber is a smooth plane curve. As a corollary, we prove a "strong" Durfee-type inequality for isolated hypersurface surface singularities, which implies Durfee's strong conjecture for such singularities with non-negative topological Euler number of the exceptional set of the minimal resolution.
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Slope equality of non-hyperelliptic Eisenbud--Harris special fibrations of genus $4$
The Horikawa index and the local signature are introduced for relatively minimal fibered surfaces whose general fiber is a non-hyperelliptic curve of genus $4$ with unique trigonal structure.
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Strange examples of local signatures for fibered surfaces of small genus
We give examples of local signatures, completely different from the usual ones, for general fibrations of genus $2$ and genus $3$.
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Upper bounds on the slope of certain fibered surfaces
We establish the slope equality and give an upper bound of the slope for finite cyclic covering fibrations of an elliptic surface including bielliptic fibrations of genus greater than 5. We also give an upper bound of the slope for triple cyclic covering fibrations of a ruled surface and hyperelliptic fibrations, which provides a new proof of Xiao's upper bound.
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Slopes of Fibered Surfaces with a Finite Cyclic Automorphism
We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.
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Fibers of Cyclic Covering Fibrations of a Ruled Surface
We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled surface and show that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. As the second application, we classify all fibers of hyperelliptic fibrations of genus 3 into 12 types according to the Horikawa index. We also prove that finite cyclic covering fibrations of a ruled surface have no multiple fibers if the degree of the covering is greater than 3.
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