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LOG CANONICAL THRESHOLDS OF DEL PEZZO SURFACES
, 2008
"... dedicated to Yuri Manin on his seventieth birthday ..."
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Cited by 41 (24 self)
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dedicated to Yuri Manin on his seventieth birthday
REFLECTION GROUPS IN ALGEBRAIC GEOMETRY
, 2008
"... After a brief exposition of the theory of discrete reflection groups in spherical, euclidean and hyperbolic geometry as well as their analogs in complex spaces, we present a survey of appearances of these groups in various areas of algebraic geometry. ..."
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Cited by 27 (3 self)
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After a brief exposition of the theory of discrete reflection groups in spherical, euclidean and hyperbolic geometry as well as their analogs in complex spaces, we present a survey of appearances of these groups in various areas of algebraic geometry.
A ThreeGeneration CalabiYau Manifold with
 Small Hodge Numbers,” Fortsch. Phys
"... We present a complete intersection CalabiYau manifold Y that has Euler number −72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z12 and the nonAbelian dicyclic group Dic3. The quotient manifolds have χ = −6 and Hodge numbers (h11, h21) = (1, ..."
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We present a complete intersection CalabiYau manifold Y that has Euler number −72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z12 and the nonAbelian dicyclic group Dic3. The quotient manifolds have χ = −6 and Hodge numbers (h11, h21) = (1, 4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the nonAbelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h11, h21) = (2, 2) that lies right at the tip of the distribution of Calabi–Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev. ar X iv
ESSENTIAL DIMENSION
"... Abstract. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions [BuR97]. Essential dimension has since b ..."
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Abstract. Informally speaking, the essential dimension of an algebraic object is the minimal number of independent parameters one needs to define it. This notion was initially introduced in the context where the objects in question are finite field extensions [BuR97]. Essential dimension has since been investigated in several broader contexts, by a range of techniques, and has been found to have interesting and surprising connections to many problems in algebra and algebraic geometry. The goal of this paper is to survey some of this research. I have tried to explain the underlying ideas informally through motivational remarks, examples and proof outlines (often in special cases, where the argument is more transparent), referring an interested reader to the literature for a more detailed treatment. The sections are arranged in rough chronological order, from the definition of essential dimension to open problems. 1. Definition of essential dimension
ON ELEMENTS OF PRIME ORDER IN THE PLANE Cremona Group Over A Perfect Field
, 2008
"... We show that the plane Cremona group over a perfect field k of characteristic p ≥ 0 contains an element of prime order ℓ ≥ 7 not equal to p if and only if there exists a 2dimensional algebraic torus T over k such that T(k) contains an element of order ℓ. If p = 0 and k does not contain a primitiv ..."
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Cited by 10 (1 self)
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We show that the plane Cremona group over a perfect field k of characteristic p ≥ 0 contains an element of prime order ℓ ≥ 7 not equal to p if and only if there exists a 2dimensional algebraic torus T over k such that T(k) contains an element of order ℓ. If p = 0 and k does not contain a primitive ℓth root of unity, we show that there are no elements of prime order ℓ> 7 in Cr2(k) and all elements of order 7 are conjugate.
VERSALITY OF ALGEBRAIC GROUP ACTIONS AND RATIONAL POINTS ON TWISTED VARIETIES
"... Abstract. We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of X (existence of rational ..."
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Abstract. We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational points on twisted forms of X (existence of rational points, existence of a dense set of rational points, etc.) Wegive applications ofthisequivalenceinbothdirections, tostudy versality of group actions and rational points on algebraic varieties. We obtain similar results on pversality for a prime integer p. An appendix, by J.P. Serre, puts the notion of versality in a historical perspective. 1.
pelementary subgroups of the Cremona group of rank 3
"... Abstract. For the subgroups of the Cremona group Cr3(C) having the form (µ p) s, where p is prime, we obtain an upper bound for s. Our bound is sharp if p ≥ 17. 1. ..."
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Abstract. For the subgroups of the Cremona group Cr3(C) having the form (µ p) s, where p is prime, we obtain an upper bound for s. Our bound is sharp if p ≥ 17. 1.