The minimal volume of stable surfaces of rank one is determined with uniqueness up to isomorphism, resolving a conjecture of Alexeev and the second author.
Rank one foliations on toroidal varieties
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abstract
Consider a log canonical pair $(X,B)$ such that there is a Cartier divisor $D$ for which $T_X(-\log B) \otimes \mathcal O(D)$ is locally free and globally generated. Let $\mathcal F$ be a log canonical foliation of rank 1 on $X$. We prove that there exists a divisor $\Gamma$ such that $(X, \Gamma)$ is log canonical and $K_X + \Gamma \sim K_{\mathcal F} + D$. We then apply this result to prove several statements on the birational geometry of rank 1 log canonical foliations on log homogeneous varieties.
fields
math.AG 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.
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The minimal volume of stable surfaces of rank one
The minimal volume of stable surfaces of rank one is determined with uniqueness up to isomorphism, resolving a conjecture of Alexeev and the second author.
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Birational boundedness of stable families
Algebraically integrable foliations of fixed dimension and bounded adjoint volume are log birationally bounded, which implies birational boundedness for stable families of maximal variation.