Title: Cremona equivalence and log Kodaira dimension Show Chinese title

Title: 克雷莫纳等价与对数卡达伊维度

Abstract: Two projective varieties are said to be Cremona equivalent if there is a Cremona modification sending one onto the other. In the last decade, Cremona equivalence has been investigated widely, and we now have a complete theory for non-divisorial reduced schemes. The case of irreducible divisors is completely different, and not much is known besides the case of plane curves and a few classes of surfaces. In particular, for plane curves it is a classical result that an irreducible plane curve is Cremona equivalent to a line if and only if its log-Kodaira dimension is negative. This can be interpreted as the log version of Castelnuovo's rationality criterion for surfaces. One expects that a similar result for surfaces in projective space should not be true, as it is false, the generalization in higher dimensions of Castelnuovo's Rationality Theorem. In this paper, the first example of such behaviour is provided, exhibiting a rational surface in the projective space with negative log-Kodaira dimension, which is not Cremona equivalent to a plane. This can be thought of as a sort of log Iskovkikh-Manin, Clemens-Griffith, Artin-Mumford example. Using this example, it is then possible to show that Cremona equivalence to a plane is neither open nor closed among log pairs with negative Kodaira dimension. Show Chinese abstract

Abstract: 两个射影概形被称为Cremona等价，如果存在一个Cremona变换将其中一个映射到另一个。 在过去的十年中，Cremona等价已被广泛研究，我们现在对非除子性约化概形有了完整的理论。 不可约除子的情况完全不同，除了平面曲线和一些曲面类外，了解甚少。 特别是对于平面曲线，这是一个经典结果：一个不可约平面曲线与一条直线Cremona等价当且仅当其对数Kodaira维数为负。 这可以解释为对表面的Castelnuovo有理准则的对数版本。 人们期望在射影空间中的曲面类似的结果不成立，因为这是Castelnuovo有理定理在高维中的推广，而该推广是不正确的。 在本文中，提供了这种行为的第一个例子，展示了一个射影空间中的有理曲面，其对数Kodaira维数为负，但并不与平面Cremona等价。 这可以看作是一种对数Iskovkikh-Manin、Clemens-Griffith、Artin-Mumford的例子。 利用这个例子，可以证明在具有负Kodaira维数的对数对中，与平面的Cremona等价既不是开的也不是闭的。