Birational geometry of nonrational rationally connected varieties

Q-Fano簇是(Log)极小模型程序的最终结果之一，是代数簇分类理论中的重要对象。我们研究了高维Q-Fano簇的双调无界性。出生无界性意味着所关注的变种形成无限多个家庭，即使我们识别出彼此出生的变种。我们证明了q-Fano簇在每一维上的二元无界性至少为3，并进一步证明了有理连通Mori纤维空间的类似结果。

Q-Fano varieties appear as one of the final outcomes of the (log) minimal model program and they are important objects in the classification theory of algebraic varieties. We studied the birational unboundedness of higher-dimensional Q-Fano varieties. The birational unboundedness means that varieties in concern form infinitely many families even if we identify varieties which are birational to each other. We have proved the birational unboundedness of Q-Fano varieties in each dimension at least 3. We further proved a similar result for rationally connected Mori fiber spaces.