𝔸-curves on log smooth varieties

In this paper, we study {\mathbb{A}^{1}}-connected varieties from log geometry point of view, and prove a criterion for {\mathbb{A}^{1}}-connectedness. As applications, we provide many interesting examples of {\mathbb{A}^{1}}-connected varieties in the case of complements of ample divisors, and the case of homogeneous spaces. We also obtain a logarithmic version of Hartshorne conjecture characterizing projective spaces and affine spaces.

CHARACTERIZATION OF PROJECTIVE SPACES AND -BUNDLES AS AMPLE DIVISORS

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_{X}$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^{n}$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^{r}$-bundle as an ample divisor by the recent work of Litt.

Higher-Level Conformal Blocks Divisors on

We study a family of semi-ample divisors on the moduli space of n-pointed genus 0 curves given by higher-level conformal blocks. We derive formulae for their intersections with a basis of 1-cycles, show that they form a basis for the Sn-invariant Picard group, and generate a full-dimensional subcone of the Sn-invariant nef cone. We find their position in the nef cone and study their associated morphisms.