Topics in Birational Geometry of Algebraic Varieties

Abstract

Various questions related to birational properties of algebraic varieties are concerned.

Rationally connected varieties are recognized as the buildings blocks of all varieties by the Minimal Model theory. We prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. As a consequence, we solve a question of Professor Burt Totaro on integral Hodge classes on rationally connected 3-folds. And by a result of Professor Claire Voisin,

the general case will be a consequence of the Tate conjecture for surfaces over finite fields.

Using the same philosophy looking for degenerated rational components through forgetful maps between moduli spaces of curves, we prove Weak Approximation conjecture to Prof. Hassett and Prof. Tschinkel for isotrivial families of rationally connected varieties. Theory of Twisted Stable maps is essentially used, with an alternative proof where some notion from Derived Algebraic Geometry is applied. It is remarkable that technics and ideas developed in this part, shed light upon and

essentially led to the final solution to weak approximation of Cubic Surfaces, which is a problem concerned by Number Theorists for many years, and this is currently the best known result in this subject.

Then we turn to Minimal Model theory in both zero and positive characteristics. Firstly, projective globally F-regular threefolds of characteristic p ≥ 11, are shown to be rationally chain connected, and back to characteristic zero, we use hard-core technics of Minimal Model program, esp. finite generate of canonical rings due to Professor Hacon, Professor McKernan et al. to characterize Toric varieties and geometric rational varieties as log canonical log-Calabi Yau varieties with "large" boundary, where the specific meanings of "large" are originated from some notion of

"charges" from String theory, and hence is related to Mirror Symmetry. This part of works also answered a Conjecture due to Prof. Shokurov.