Maps with Prescribed Boardman Singularities

Doklady Mathematics - In this paper we extend Y. Eliashberg’s theorem on the maps with fold type singularities to arbitrary Thom-Boardman singularities. Namely, we state a necessary and...     

Hypersurfaces with too many rational curves

We study smooth hypersurfaces of degree \(d\ge n+1\) in \(\mathbf{P}^n\) whose spaces of smooth rational curves of low degrees are larger than expected, and show that under certain conditions, the primitive part of the middle cohomology of such hypersurfaces have non-trivial Hodge substructures. As an application, we prove that the space of lines on any smooth Fano hypersurface of degree \(d \le 8\) in \(\mathbf{P}^n\) has the expected dimension \(2n-d-3\) .     

Analytic Representation of CR Functions on Hypersurfaces with Singularities

We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.     

Starshaped Locally Convex Hypersurfaces with Prescribed Curvature and Boundary

In this paper we find strictly locally convex hypersurfaces in \(\mathbb {R}^{n+1}\) with prescribed curvature and boundary. The main result is that if the given data admits a strictly locally convex radial graph as a subsolution, we can find a radial graph realizing the prescribed curvature and boundary. As an application we show that any smooth domain on the boundary of a compact strictly convex body can be deformed to a smooth hypersurface with the same boundary (inside the convex body) and realizing any prescribed curvature function smaller than the curvature of the body.     

Levi-Flat Hypersurfaces Tangent to Projective Foliations

Let M?? ⁿ be a singular real-analytic Levi-flat hypersurface tangent to a codimension-one holomorphic foliation \(\mathcal{F}\) on ? ⁿ . For n≥3, we give sufficient conditions to guarantee the existence of degenerate singularities in M, (in the sense of Segre varieties) and as a consequence we prove that \(\mathcal{F}\) can be defined by a global closed meromorphic 1-form.     

Projective surfaces with many skew lines

We give an example of a smooth surface of degree that contains pairwise disjoint lines. In particular, our example shows that the degree in Miyaoka's bound is sharp.

     

Convex Hypersurfaces with Prescribed Curvature and Boundary in Riemannian Manifolds

In this article a priori estimates at the boundary for the second fundamental form of n-dimensional convex hypersurfaces M with prescribed curvature quotient Sn (M)/Sl (M) in Riemannian manifolds are derived. A consequence of these estimates and other known results is an existence theorem for such hypersurfaces, which is a generalization of a recent result of Ivochkina and Tomi to the Riemannian case.     

Hypersurfaces and Their Singularities in Partial Correlation Testing

An asymptotic theory is developed for computing volumes of regions in the parameter space of a directed Gaussian graphical model that are obtained by bounding partial correlations. We study these volumes using the method of real log canonical thresholds from algebraic geometry. Our analysis involves the computation of the singular loci of correlation hypersurfaces. Statistical applications include the strong-faithfulness assumption for the PC algorithm and the quantification of confounder bias in causal inference. A detailed analysis is presented for trees, bow ties, tripartite graphs, and complete graphs.     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)     

Starshaped Compact Hypersurfaces with Prescribed M-th Mean Curvature in Elliptic Space

We consider the problem of finding a compact starshaped hypresurface in a space form for which the normalized m-th elementary symmetric function of principal curvatures is a prescribed function. In this paper the conditions for the existence of at least one solution to a nonlinear second order elliptic equation of that problem are established in case of a space form with positive sectional curvature.     

Real Hypersurfaces with Two Principal Curvatures in Complex Projective and Hyperbolic Planes

We find the first examples of real hypersurfaces with two nonconstant principal curvatures in complex projective and hyperbolic planes, and we classify them. It turns out that each such hypersurface is foliated by equidistant Lagrangian flat surfaces with parallel mean curvature or, equivalently, by principal orbits of a cohomogeneity two polar action.     

复射影空间的实2-调和超曲面

本文研究了复射影空间中的实２－调和超曲面和实极小超曲面之间的关系,推广了Ｌａｗｓｏｎ Ｈ．Ｂ和Ｋｏｎ Ｍ．关于实极小超曲面的Ｐｉｎｃｈｉｎｇ结果     

The Varieties of Tangent Lines to Hypersurfaces in Projective Spaces

For a hypersurface in a projective space, we consider the set of pairs of a point and a line in the projective space such that the line intersects the hypersurface at the point with a fixed multiplicity. We prove that this set of pairs forms a smooth variety for a general hypersurface.     

射影空间中代数曲线与超曲面的一些新的性质

发现了代数曲线的新的不变量一特征数,并得到了Pascal定理的不同于3次曲线的Cllasles定理和高次曲线中的Cayley-Bacharach定理等形式的高次推广.进一步研究了平面代数曲线的一些性质.通过定义m次Pascal超曲面,将Pascal定理推广到n维射影空间的m次超曲面中,证明了n-单纯形上的Pascal点位于一个m次Pascal超曲面的充要条件是其每个2维面上的Pascal点分别位于m次平面Pascal空间的一条代数曲线上.进一步,给出了一定条件下m次Pascal超曲面与m-1次Pascal超曲面之间的内在关系.     

广义de Sitter空间中法向Lorentzian超曲面的nullcone Gauss映射的奇点

利用奇点理论研究了广义de Sitter空间中具有Lorentzian法空间的一类超曲面.介绍了这类超曲面的局部微分几何,定义了nullcone Gauss映射及nullcone高度函数族,进而研究了nullcone高度函数族的性质及nullcone高斯映射的几何意义,最后研究了这类超曲面的通有性质.