Preprint Download

Back to Home Page of Urabe Lab

List of available preprint

1. The Mathematica programs used in "Calculation of Manin's invariant for Del Pezzo surfaces" (Mathemarics of computation 65, No. 213, 247-258 and S15-S23) (1993, BinHex, 224KB)

2. The Gauss map and the dual variety of real-analytic submanifolds in a sphere or in hyperbolic space (1995, revised in 1997 pdf, 416KB)

   Abstract: We study the Gauss map and the dual variety of a real-analytic immersion of a connected compact real-analytic manifold into a sphere or into hyperbolic space. The dual variety is defined to be the set of all normal directions of the immersion. First, we show that the image of the Gauss map characterizes the manifold. Also we show that the dual variety characterizes the manifold. Besides, duality of the second fundamental form and some results on degeneration are obtained.

   Keywords: Gauss map, dual variety, second fundamental form
   Subject class: Primary 53A20, Secondary 14P15

3. Duality of the second fundamental form (1997, pdf, 96KB)

   Abstract: First, we consider a compact real-analytic irreducible subvariety $M$ in a sphere and its dual variety $M^\vee$. We explain that two matrices of the second fundamental forms for both varieties $M$ and $M^\vee$ can be regarded as the inverse matrices of each other. Also generalization in hyperbolic space is explained.

   Keywords: dual variety, second fundamental form
   Subject class: Primary 51N20, Secondary 32C05

4. Some fundamental problems on real-analytic sets (1997, pdf, 96KB)

   Abstract: The category of real-analytic sets and real-analytic maps is the most important category in application. However, in spite of efforts by F. Bruhat, H. Cartan, H. Whitney et al., the basic theory of real-analytic category does not yet seem to be well-developed. In this article I would like to point out several basic problems.

   Keywords: real-analytic, smooth, irreducible
   Subject class: Primary 32C07, Secondary 26-02

5. On weak Hironaka Theorem-Part 2 (1997, pdf, 256KB, in Japanese)

   Abstract: Some review of recent developments of theory of resolution of singularities of algebraic varieties.

   Keywords: resolution, singularity, normal crossing
   Subject class: Primary 14E15, Secondary 32S45

6. Resolution of singularities of germs in characteristic positive associated with valuation rings of iterated divisor type(1999, pdf, 672KB)

   Abstract: In this paper we show that any hypersurface singularities of germs of varieties in positive characteristic can be resolved by iterated monoidal transformations in centers in smooth subvarieties, if we have a valuation ring of iterated divisor type associated with the germ. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transformations with a normal crossing, Weierstrass representations, reduction sequences, and so forth.

   Keywords: singularity, resolution of singularities, monoidal transformation, normal crossing, Weierstrass preparation theorem, characteristic positive, valuation
   Subject class: Primary 14B05, Secondary 32S05, 13H99

7. Hypersurface Singularities which cannot be resolved in Characteristic Positive(1999, pdf, 652KB)

   Abstract: Assume that there exists a hypersurface singularity which cannot be resolved by iterated monoidal transformations in positive characteristic. We show that in the set of defining functions of hypersurface singularities which cannot be resolved, we can find a function satisfying very strong conditions. By these conditions we may be able to deduce a contradiction under the above assumption. Besides, we introduce fundamental concepts for the study of resolution of singularities of germs such as space germs, iterated analytic monoidal transforms with a normal crossing, Weierstrass representations, reduction sequences, and so forth. This is a revised version of a part of contents of my previous manuscript "Resolution of Singularities of Germs in Characteristic Positive associated with Valuation Rings of Iterated Divisor Type" at http://xxx.lanl.gov/abs/math.AG/9901048.

   Keywords: hypersurface, singularity, resolution of singularities, monoidal transformation, normal crossing, Weierstrass preparation theorem, positive characteristic
   Subject class: Primary 14B05, Secondary 32S05, 13H99

8. New ideas for resolution of singularities in artitrary characteristic(2000, pdf, 232KB)

   Abstract: I have succeeded in showing that any two-dimensional hypersurface singularities of germs of varieties in any characteristic can be resolved by iterated monoidal transformations with centers in smooth subvarieties. The new proof for the two-dimensional case depends on new ideas. Ideas are essentially different from Abhyankar's one in 1956 and Lipman's one in 1978. It seems to be possible to generalize the new proof into higher dimensional cases, if we add several ideas further. In this article I try to explain my new ideas.

   Keywords: hypersurface, singularity, resolution of singularities, monoidal transformation, blowing-up, normal crossing, Weierstrass preparation theorem
   Subject class: Primary 14B05; Secondary 32S05, 13F25

Back to Home Page of Urabe Lab

Send your opinion to .