On the intersection of rational transversal subtori

Elementary Transversality in the Schubert calculus in any Characteristic

We give a characteristic-free proof that general codimension-1 Schubert varieties meet transversally in a Grassmannian and in some related varieties. Thus the corresponding intersection numbers computed in the Chow (and quantum Chow) rings of these varieties are enumerative in all characteristics. We show that known transversality results do not apply to these enumerative problems, emphasizing the need for additional theoretical work on transversality. The method of proof also strengthens some results in real enumerative geometry.

关于亏格3超椭圆轨迹与超奇异轨迹的非横截性

On the non-Transversality of the Hyperelliptic Locus and the Supersingular Locus for $g=3$

本文研究亏格3时西格尔模叠中超椭圆轨迹与超奇异轨迹的交点性质，给出非横截交点的判别准则，并证明对无穷多个素数p存在此类交点。

Tropical complete intersection curves

A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the genus of C (defined as the number of independent cycles of C) when C is smooth and connected.

Chern classes of splayed intersections

We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a 'splayedness' assumption. The relation is shown to hold for both the Chern-Schwartz-MacPherson class and the Chern-Fulton class. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Geometry of intersections of some secant varieties to algebraic curves

For a smooth projective curve, the cycles of subordinate or, more generally, secant divisors to a given linear series are among some of the most studied objects in classical enumerative geometry. We consider the intersection of two such cycles corresponding to secant divisors of two different linear series on the same curve and investigate the validity of the enumerative formulas counting the number of divisors in the intersection. We study some interesting cases, with unexpected transversality properties, and establish a general method to verify when this intersection is empty.

Discriminant of the ordinary transversal singularity type. The local aspects

Consider a space X with the singular locus, Z=Sing(X), of positive dimension. Suppose both Z and X are locally complete intersections. The transversal type of X along Z is generically constant but at some points of Z it degenerates. We introduce (under certain conditions) the discriminant of the transversal type, a subscheme of Z, that reflects these degenerations whenever the generic transversal type is `ordinary'. The scheme structure of this discriminant is imposed by various compatibility properties and is often non-reduced. We establish the basic properties of this discriminant: it is a C…

Towards transversality of singular varieties: splayed divisors

We study a natural generalization of transversally intersecting smooth hypersurfaces in a complex manifold: hypersurfaces, whose components intersect in a transversal way but may be themselves singular. Such hypersurfaces will be called splayed divisors. A splayed divisor is characterized by a property of its Jacobian ideal. This yields an effective test for splayedness. Two further characterizations of a splayed divisor are shown, one reflecting the geometry of the intersection of its components, the other one using K. Saito's logarithmic derivations. As an application we prove that a union o…

Pursuing the double affine Grassmannian I: transversal slices via instantons on A_k-singularities

This paper is the first in a series that describe a conjectural analog of the geometric Satake isomorphism for an affine Kac-Moody group. In this paper we construct a model for the singularities of some would-be Schubert varieties in the affine Grassmannian for an affine Kac-Moody group. We formulate a conjecture describing the (local) intersection cohomology of these varieties in terms of integrable representations of the Langlands dual affine Kac-Moody group and check this conjecture in a number of cases. Roughly speaking the above singularities are constructed by looking at the Uhlenbeck sp…

On the Chern classes of singular complete intersections

We consider two classical extensions for singular varieties of the usual Chern classes of complex manifolds, namely the total Schwartz-MacPherson and Fulton-Johnson classes, $c^{SM}(X)$ and $c^{FJ}(X)$. Their difference (up to sign) is the total Milnor class ${\mathcal M}(X)$, a generalization of the Milnor number for varieties with arbitrary singular set. We get first Verdier-Riemann-Roch type formulae for the total classes $c^{SM}(X)$ and $c^{FJ}(X)$, and use these to prove a surprisingly simple formula for the total Milnor class when $X$ is defined by a finite number of local complete inter…

仿射球面质量与热带双扇的高斯运动学公式

Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans

本文在秩四格中引入仿射球面质量并证明其与稳定自交和最小次数的双边不等式，结合高斯-运动学公式与仿射规范化原理，给出四维横截性估计及协变性，并探讨最优常数问题。

Vanishing cycles on Poisson varieties

We extend slightly the results of Evens-Mirković, and "compute" the characteristic cycles of Intersection Cohomology sheaves on the transversal slices in the double affine Grassmannian and on the hypertoric varieties. We propose a conjecture relating the hyperbolic stalks and the microlocalization at a torus-fixed point in a Poisson variety.

Singularities of generalized Richardson varieties

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this paper, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singula…