We prove that there is a first-order sentence in the language of rings that is true for all finitely generated fields of characteristic $0$ and false for all fields of characteristic greater than $0$. We also prove that for each $n\in \mathbb{N}$, there is a first-order formula ${\psi }_n(x_1,\dots ,x_n)$ that when interpreted in a finitely generated field $K$ is true for elements $x_1,\dots ,x_n\in K$ if and only if the elements are algebraically dependent over the prime field in $K$

We study moduli of semistable twisted sheaves on smooth proper algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asymptotically geometrically irreducible, normal, generically smooth, and locally complete intersections (l.c.i.'s) over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, twisted Bogomolov inequalities, semistability and boundedness results, and basic results on twisted $\mathrm{Quot}$-schemes on a surface

We prove the rational connectedness conjecture of V. V. Shokurov in [20] which, in particular, implies that the fibres of a resolution of a variety with divisorial log terminal singularities are rationally chain connected

Let $G_{n,k}(\mathbb{K})$ be the Grassmannian manifold of $k$-dimensional $\mathbb{K}$-subspaces in ${\mathbb{K}}^n$, where $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$ is the field of real, complex, or quaternionic numbers. For $1\le k<k^{\text{'}}\le n-1$, we define the Radon transform $(Rf)(\eta )$, $\eta \in G_{n,k^{\text{'}}}(\mathbb{K})$, for functions $f(\xi )$ on $G_{n,k}(\mathbb{K})$ as an integration over all $\xi \subset \eta$. When $k+k^{\text{'}}\le n$, we give an inversion formula in terms of the Gårding-Gindikin fractional integration and the Cayley-type differential operator on the symmetric cone of positive ($k\times k$)-matrices over $\mathbb{K}$. This generalizes the recent results of Grinberg and Rubin [4] for real Grassmannians

The extended Toda hierarchy (ETH) was introduced by E. Getzler [Ge] and independently by Y. Zhang [Z] in order to describe an integrable hierarchy that governs the Gromov-Witten invariants of $\mathbb{C}{P}^1$. The Lax-type presentation of the ETH was given in [CDZ]. In this article, we give a description of the ETH in terms of tau functions and Hirota quadratic equations (HQEs), also known as Hirota bilinear equations (HBEs). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line