Connectedness Bertini Theorem via numerical equivalence

实闭域上的广义连通性与Bertini型定理

Generalized connectedness and Bertini-type theorems over real closed fields

本文建立了Bertini定理在实闭域上的类比，证明了可逆层截面的零轨迹若具形式实泛点则不变号，并证明维数≥2的拟射影簇存在保持形式实性与整性的超曲面截面非空开子集。

Small codimension subvarieties in homogeneous spaces

We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of $X$. Finally we prove transplanting theorems {à} la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.

R-equivalence and A^1-connectedness in anisotropic groups

We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A^1-equivalent. As a consequence, we see that Sing_*(G) cannot be A^1-local for such groups. This implies that the A^1-connected components of a semisimple, absolutely almost simple, simply connected group over a field k form a sheaf of abelian groups.

Fibration by non-smooth projective curves of arithmetic genus two in characteristic two

Looking in positive characteristic for failures of the Bertini-Sard theorem, we determine, up to birational equivalence, the separable proper morphisms of smooth algebraic varieties in characteristic two, whose fibres are non-smooth curves of arithmetic genus two.

论有限伽罗瓦扩张的有理连通性与参数化

On rational connectedness and parametrization of finite Galois extensions

该研究围绕有理数域的$G$-伽罗瓦扩张的$R$-等价关系展开，确定了基本群族$G$的$R$-等价类，在$G$无通用扩张的情况下实现了对应伽罗瓦扩张的参数化。

在特征零域上，单有理性与有理连通性是等价的

Unirationality is the Same as Rational Connectedness in Characteristic Zero

本文通过构造类似 MRC 纤维化的 MU 纤维化，证明了特征零域上光滑簇的单有理性、有理连通性与有理链连通性等价，并指出 MRC 商非单有理覆盖。