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Register a new userSpecial birational transformations of type (2,1)
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A birational transformation f: P^n --> Z, where Z is a nonsingular variety of Picard number 1, is called a special birational transformation of type (a, b) if f is given by a linear system of degree a, its inverse is given by a linear system of degree b and the base locus S subset P^n of f is irreducible and nonsingular. In this paper, we classify special birational transformations of type (2,1). In addition to previous works Alzati-Sierra and Russo on this topic, our proof employs natural C^*-actions on Z in a crucial way. These C^*-actions also relate our result to the problem studied in our previous work on smooth projective varieties with nonzero prolongations.
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We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.
We classify simple groups that act by birational transformations on compact complex Kahler surfaces. Moreover, we show that every finitely generated simple group that acts non-trivially by birational transformations on a projective surface over an arbitrary field is finite.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V)>0$ and $P_{24}(V)>1$ (which answers an open problem of J. Kollar and S. Mori). We also prove that the canonical volume has an universal lower bound $text{Vol}(V) geq 1/2660$ and that the pluri-canonical map $Phi_m$ is birational onto its image for all $mgeq 77$. As an application of our method, we prove Fletchers conjecture on weighted hyper-surface 3-folds with terminal quotient singularities. Another featured result is the optimal lower bound $text{Vol}(V)geq {1/420}$ among all those 3-folds $V$ with $chi({mathcal O}_V)leq 1$.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0leq 24$. A direct consequence is the birationality of the pluricanonical map $varphi_m$ for all $mgeq 126$. Besides, the canonical volume $text{Vol}(V)$ has a universal lower bound $ u(3)geq frac{1}{63cdot 126^2}$.
Let $V$ be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of $V$. We show that the $m$-canonical map of $V$ is birational for all $mgeq 73$ and that the canonical volume $text{Vol}(V)geq {1/2660}$. When $chi(mathcal{O}_V)leq 1$, our result is $text{Vol}(V)geq {1/420}$, which is optimal. Other effective results are also included in the paper.
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