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It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Venereau established that v-n is indeed a variable of C[x][y,z,u] over C[x] for n>= 3. In this note we give an alternative proof of Venereau's... more
It is an open question whether v1 and v2 are variables of the polynomial ring C[x,y,z,u]. S. Venereau established that v-n is indeed a variable of C[x][y,z,u] over C[x] for n>= 3. In this note we give an alternative proof of Venereau's result based on the above equivalence. We also discuss some other equivalent properties, as well as the relations to the Abhyankar-Sathaye Embedding Problem and to the Dolgachev-Weisfeiler Conjecture on triviality of flat families with fibers affine spaces.
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This report records a large number of open problems in Affine Algebraic Geometry that were proposed by participants in a Conference on Open Algebraic Varieties at the Centre de Recherches en Mathematiques in Montreal at December 1994.
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... In noncompact variants, statements appear which resemble function-field analogs of the Siegel-Mahler theorem on integral points. For families of elliptic curves they are established in [1] (Theorem 3) and [2] (the corollary to Theorem... more
... In noncompact variants, statements appear which resemble function-field analogs of the Siegel-Mahler theorem on integral points. For families of elliptic curves they are established in [1] (Theorem 3) and [2] (the corollary to Theorem 4); for fam-ilies of abelian varieties over C in ...
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Uniqueness of % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX % garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz % aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq % Fr0xc9pk0xbba9q8WqFfea0-yr0RY...more
A Gizatullin surface is a normal affine surface V over , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of -actions and -fibrations on such a... more
A Gizatullin surface is a normal affine surface V over , which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of -actions and -fibrations on such a surface V up to automorphisms. The latter fibrations are in one to one correspondence with
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61p. v. aussi : V. Lin and M. Zaidenberg, Configuration spaces of the affine line and their automorphism groups In: Automorphisms in Birational and Complex Geometry. Ivan Cheltsov et al. (eds.), 431-468. Springer Proceedings in... more
61p. v. aussi : V. Lin and M. Zaidenberg, Configuration spaces of the affine line and their automorphism groups In: Automorphisms in Birational and Complex Geometry. Ivan Cheltsov et al. (eds.), 431-468. Springer Proceedings in Mathematics and Statistics, vol. 79, 2014.
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In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d\geqslant 5$ in $\PP^ 3$ (the cases $d\leqslant 4$ are well known). we introduce the set... more
In this paper we consider the question of determining the geometric genera of irreducible curves lying on a very general surface $S$ of degree $d\geqslant 5$ in $\PP^ 3$ (the cases $d\leqslant 4$ are well known). we introduce the set ${\rm Gaps}(d)$ of all non--negative integers which are not realized as geometric genera of irreducible curves on a very general surface of degree $d$ in $\PP^ 3$. We prove that ${\rm Gaps}(d)$ is finite and, in particular, that ${\rm Gaps}(5)= \{0,1,2\}$. The set ${\rm Gaps}(d)$ is the union of finitely many disjoint and separated integer intervals. The first of them, according to a theorem of Xu, is ${\rm Gaps}_0(d):=\left[0, \; \frac{d(d-3)}{2} - 3\right]$. We show that the next one is ${\rm Gaps}_1(d):=\left[\frac{d^2-3d+4}{2}, \; d^2 - 2d - 9\right]$ for all $d\geqslant 6$.
Given a normal affine surface X with a nontrivial ℂ * -action, the coordinate algebra A=ℂ[X] comes equipped with a ℤ-grading. There are three basic types of gradings and ℂ * -actions: A 0 =ℂ and A + or A - =0 (elliptic case); A 0 ≠ℂ and A... more
Given a normal affine surface X with a nontrivial ℂ * -action, the coordinate algebra A=ℂ[X] comes equipped with a ℤ-grading. There are three basic types of gradings and ℂ * -actions: A 0 =ℂ and A + or A - =0 (elliptic case); A 0 ≠ℂ and A + or A - =0 (parabolic case); A ± ≠0 (hyperbolic case). H. Pinkham [Math. Ann. 227, 183–193 (1977; Zbl 0338.14010)] and M. Demazure [Trav. Cours 37, 35–68 (1988; Zbl 0686.14005)] proved that in elliptic and parabolic case A=⨁ n≥0 H 0 (C,[nD]) for a certain projective or affine curve C and a ℚ-divisor D on C, and the graded isomorphism class of A is determined by the rational equivalence class of D. In the parabolic case, the Pinkham–Demazure construction is used in the paper to show that each rational parabolic ℂ * -surface is realized as the normalization of a surface in 𝔸 3 given by an equation x d =P(z)y. Also, the Pinkham–Demazure construction is generalized in the paper to hyperbolic case in the following form: coordinate algebras of hyperboli...