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On Hrushovski’s proof of the ManinMumford conjecture
 in Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 539– 546, Higher Ed. Press, Beijing, 2002. TETSUSHI ITO
"... The ManinMumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening ..."
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The ManinMumford conjecture in characteristic zero was first proved by Raynaud. Later, Hrushovski gave a different proof using model theory. His main result from model theory, when applied to abelian varieties, can be rephrased in terms of algebraic geometry. In this paper we prove that intervening result using classical algebraic geometry alone. Altogether, this yields a new proof of the ManinMumford conjecture using only classical algebraic geometry.
The group of automorphisms of a real rational surface is ntransitive
 arXiv:0708.3992 [math.AG] JÉRÉMY BLANC AND FRÉDÉRIC MANGOLTE
"... Abstract. Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts ntransitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that ..."
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Cited by 10 (6 self)
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Abstract. Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts ntransitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.
Antiaffine algebraic groups
 J. Algebra
"... Abstract. We say that an algebraic group G over a field is antiaffine if every regular function on G is constant. We obtain a classification of such groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert’ ..."
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Cited by 7 (5 self)
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Abstract. We say that an algebraic group G over a field is antiaffine if every regular function on G is constant. We obtain a classification of such groups, with applications to the structure of algebraic groups in positive characteristics, and to the construction of many counterexamples to Hilbert’s fourteenth problem. In this article, we introduce and study the class of groups of the title. We say that a group scheme G of finite type over a field k is antiaffine if O(G) = k; then G is known to be connected, commutative and smooth. Examples include abelian varieties, their universal vector extensions (in
Some basic results on actions of nonaffine algebraic groups, arXiv: math.AG/0702518, to appear in the proceedings of the conference “Symmetry and Spaces
"... Abstract. We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. This yields a structure theorem for normal equivariant embeddings of semiabelian varieties, and a characteristicfree version of the Borel–Remmert theo ..."
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Cited by 6 (6 self)
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Abstract. We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. This yields a structure theorem for normal equivariant embeddings of semiabelian varieties, and a characteristicfree version of the Borel–Remmert theorem. Algebraic group actions have been extensively studied under the assumption that the acting group is affine or, equivalently, linear; see [15, 19, 21]. In contrast, little seems to be known about actions of nonaffine algebraic groups. In this paper, we show that these actions
Counting points of homogeneous varieties over finite fields
, 2008
"... Abstract. Let X be an algebraic variety over a finite field Fq, homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N, the number of rational points of X over Fq n is a polynomial function of q n with in ..."
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Cited by 2 (2 self)
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Abstract. Let X be an algebraic variety over a finite field Fq, homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N, the number of rational points of X over Fq n is a polynomial function of q n with integer coefficients. Moreover, the shifted polynomials, where q n is formally replaced with q n + 1, have nonnegative coefficients.
ON THE GEOMETRY OF ALGEBRAIC GROUPS AND HOMOGENEOUS SPACES
"... Abstract. Given a connected algebraic group G over an algebraically closed field and a Ghomogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the “antiaffine ” fibrations of G and X. 1. ..."
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Cited by 1 (1 self)
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Abstract. Given a connected algebraic group G over an algebraically closed field and a Ghomogeneous space X, we describe the Chow ring of G and the rational Chow ring of X, with special attention to the Picard group. Also, we investigate the Albanese and the “antiaffine ” fibrations of G and X. 1.
LOCAL STRUCTURE OF ALGEBRAIC MONOIDS
, 709
"... Abstract. We describe the local structure of an irreducible algebraic monoid M at an idempotent element e. When e is minimal, we show that M is an induced variety over the kernel MeM (a homogeneous space) with fibre the twosided stabilizer Me (a connected affine monoid having a zero element and a d ..."
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Abstract. We describe the local structure of an irreducible algebraic monoid M at an idempotent element e. When e is minimal, we show that M is an induced variety over the kernel MeM (a homogeneous space) with fibre the twosided stabilizer Me (a connected affine monoid having a zero element and a dense unit group). This yields the irreducibility of stabilizers and centralizers of idempotents when M is normal, and criteria for normality and smoothness of an arbitrary monoid M. Also, we show that M is an induced variety over an abelian variety, with fiber a connected affine monoid having a dense unit group. An algebraic monoid is an algebraic variety equipped with an associative product map, which is a morphism of varieties and admits an identity element. Algebraic monoids are closely related to algebraic groups: the group G of invertible elements of any irreducible algebraic
F.: The group of algebraic diffeomorphisms of a real rational surface is ntransitive
"... Abstract. Let X be a rational nonsingular compact connected real algebraic surface. Denote by Diffalg(X) the group of algebraic diffeomorphisms of X into itself. The group Diffalg(X) acts diagonally on X n, for any natural integer n. We show that this action is transitive, for all n. As an applicati ..."
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Cited by 1 (1 self)
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Abstract. Let X be a rational nonsingular compact connected real algebraic surface. Denote by Diffalg(X) the group of algebraic diffeomorphisms of X into itself. The group Diffalg(X) acts diagonally on X n, for any natural integer n. We show that this action is transitive, for all n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are algebraically diffeomorphic if and only if they are homeomorphic as topological surfaces.
UNDECIDABLE PROBLEMS: A SAMPLER
"... Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence ..."
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Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. 1. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number strictly between ℵ0 and 2 ℵ0, is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [Göd40, Coh63, Coh64].) The first examples of statements independent of a “natural ” axiom system were constructed by K. Gödel [Göd31]. 2. Decision problem: A family of problems with YES/NO answers is called undecidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert’s tenth problem, to decide whether a multivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 1.1. In modern literature, the word “undecidability ” is used more commonly in sense 2, given that “independence ” adequately describes sense 1. To make 2 precise, one needs a formal notion of algorithm. Such notions were introduced by A. Church [Chu36a] and A. Turing [Tur36] independently in the 1930s. From now on, we interpret algorithm to mean Turing machine, which, loosely speaking, means that it is a computer program that takes as input a finite string of 0s and 1s. The role of the finite string is to specify which problem in the family is to be solved. Remark 1.2. Often in describing a family of problems, it is more convenient to use higherlevel mathematical objects such as polynomials or finite simplicial complexes as input. This is acceptable if these objects can be encoded as finite binary strings. It is not necessary to specify the encoding as long as it is clear that a Turing machine could convert between reasonable encodings imagined by two different readers.