Nidhi Gupta
TIFR, Mumbai
September 25, 2025
$\mathbb A^1$-connectedness of affine quadratic hypersurfaces.: It is a classical fact that a smooth projective quadratic hypersurface over a field $k$ is rational precisely when it admits a $k$-rational point. Let $k$ be a field of characteristic 0. A theorem of Asok-Morel further shows that any smooth projective rational variety is $\mathbb{A}^1$-connected, so for smooth projective quadrics over $k$, $\mathbb{A}^1$-connectedness is governed by the existence of a $k$-rational point. In this talk, I will present an analogous characterization for affine quadrics in terms of classical invariants of the associated quadratic form. The argument draws on the study of rational curves on these hypersurfaces together with techniques from quadratic form theory. This talk is based on joint work with Dr. Chetan Balwe.
Rajat Kumar Mishra
TIFR, Mumbai
September 25, 2025
Differential Characters and The Manin Kernel: Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\Spec K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ (also known as the Manin Kernel) is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$.
TIFR, Mumbai
September 25, 2025
$\mathbb A^1$-connectedness of affine quadratic hypersurfaces.: It is a classical fact that a smooth projective quadratic hypersurface over a field $k$ is rational precisely when it admits a $k$-rational point. Let $k$ be a field of characteristic 0. A theorem of Asok-Morel further shows that any smooth projective rational variety is $\mathbb{A}^1$-connected, so for smooth projective quadrics over $k$, $\mathbb{A}^1$-connectedness is governed by the existence of a $k$-rational point. In this talk, I will present an analogous characterization for affine quadrics in terms of classical invariants of the associated quadratic form. The argument draws on the study of rational curves on these hypersurfaces together with techniques from quadratic form theory. This talk is based on joint work with Dr. Chetan Balwe.
Rajat Kumar Mishra
TIFR, Mumbai
September 25, 2025
Differential Characters and The Manin Kernel: Let $K$ be a field of characteristic zero with a fixed derivation $\partial$ on it. In the case when $A$ is an abelian scheme, Buium considered the group scheme $K(A)$ which is the kernel of differential characters (also known as Manin characters) on the jet space of $A$. Then $K(A)$ naturally inherits a $D$-group scheme structure. Using the theory of universal vectorial extensions of $A$, he further showed that $K(A)$ is a finite dimensional vectorial extension of $A$. Let $G$ be a smooth connected commutative finite dimensional group scheme over $\Spec K$. In this paper, using the theory of differential characters, we show that the associated kernel group scheme $K(G)$ (also known as the Manin Kernel) is a finite dimensional $D$-group scheme that is a vectorial extension of such a general $G$.