Qile Chen

Skeletons and Fans of Logarithmic Structures

We survey a collection of closely related methods for generalizing fans of toric varieties, include skeletons, Kato fans, Artin fans, and polyhedral cone complexes, all of which apply in the wider context of logarithmic geometry. Under appropriate assumptions these structures are equivalent, but their different realizations have provided for surprisingly disparate uses. We highlight several current applications and suggest some future possibilities.

Very free curves on Fano complete intersections

In this paper, we show that general Fano complete intersections over an algebraically closed eld of arbitrary characteristic are separably rationally connected. Our construction of rational curves leads to a more interesting generalization that general log Fano complete intersections with smooth tame boundary divisors admit very free A 1 -curves.

The degeneration formula for logarithmic expanded degenerations

We define a notion of logarithmic stable maps based on Bumsig Kims construction. Then we prove a degeneration formula under this setting by applying the method develeoped by Dan Abramovich and Barbara Fantechi for transversal maps.

On the Irreducibility of the Space of Genus Zero Stable Log Maps to Wonderful Compactifications

In this paper, we prove the moduli spaces of genus zero stable log maps to a large class of wonderful compactifications are irreducible and unirational.

curves on log smooth varieties

Abstract In this paper, we study 𝔸 1 {\mathbb{A}^{1}} -connected varieties from log geometry point of view, and prove a criterion for 𝔸 1 {\mathbb{A}^{1}} -connectedness. As applications, we provide many interesting examples of 𝔸 1 {\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.