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The automorphism group of
Full text linkLet be the moduli stack parametrizing Deligne-Mumford stable -pointed genus curves and let be its coarse moduli space: the Deligne-Mumford compactification of the moduli space of -pointed genus smooth curves. We prove that the automorphism groups of and are isomorphic to the symmetric group on elements for any such that , and compute the remaining cases.22 page
Biregular and Birational Geometry of Algebraic Varieties
Full text linkEvery area of mathematics is characterized by a guiding problem. In algebraic geometry such problem is the classification of algebraic varieties. In its strongest form it means to classify varieties up to biregular morphisms. However, birationally equivalent varieties share many interesting properties. Therefore for any birational equivalence class it is natural to work out a variety, which is the simplest in a suitable sense, and then study these varieties. This is the aim of birational geometry. In the first part of this thesis we deal with the biregular geometry of moduli spaces of curves, and in particular with their biregular automorphisms. However, in doing this we will consider some aspects of their birational geometry. The second part is devoted to the birational geometry of varieties of sums of powers and to some related problems which will lead us to computational geometry and geometric complexity theory
On the biregular geometry of the Fulton–MacPherson compactification
No full textLet X[n] be the Fulton–MacPherson compactification of the configuration space of n ordered points on a smooth projective variety X. We prove that if either n≠2 or dim(X)≥2, then the connected component of the identity of Aut(X[n]) is isomorphic to the connected component of the identity of Aut(X). When X=C is a curve of genus g(C)≠1 we classify the dominant morphisms C[n]→C[r], and thanks to this we manage to compute the whole automorphism group of C[n], namely Aut(C[n])≅Sn×Aut(C) for any n≠2, while Aut(C[2])≅S2⋉(Aut(C)×Aut(C)). Furthermore, we extend these results on the automorphisms to the case where X=C1×&×Cr is a product of curves of genus g(Ci)≥2. Finally, using the techniques developed to deal with Fulton–MacPherson spaces, we study the automorphism groups of some Kontsevich moduli spaces M ̅0,n(PN,d)
Complete symplectic quadrics and Kontsevich spaces of conics in Lagrangian Grassmannians
Full text linkA wonderful compactification of an orbit under the action of a semi-simple and simply connected group is a smooth projective variety containing the orbit as a dense open subset, and where the added boundary divisor is simple normal crossing. We construct the wonderful compactification of the space of symmetric and symplectic matrices, and investigate its geometry. As an application, we describe the birational geometry of the Kontsevich spaces parametrizing conics in Lagrangian Grassmannians
The rank of n × n matrix multiplication is at least 3n2 - 2√2n3/2 - 3n
No full textWe prove that the rank of the n×n matrix multiplication is at least 3n2 - 2√2n3/2 - 3n. The previous bounds were 3 n2-4n32-n due to Landsberg [2] and 52n2-3n due to Bläser [1]. Our bound improves the previous bounds for any n≥24. © 2013 Elsevier Inc. All rights reserved
On the automorphisms of Hassett’s moduli spaces
No full textLet Mg,A[n] be the moduli stack parametrizing weighted stable curves, and let Mg,A[n] be its coarse moduli space. These spaces have been introduced by B. Hassett, as compactifications of Mg,n and Mg,n, respectively, by assigning rational weights A = (a1, ..., an), 0 < ai ≤ 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a1 = · · · = an = 1. In genus zero some of these spaces appear as intermediate steps of the blow-up construction of M0,n developed by M. Kapranov, while in higher genus they may be related to the LMMP on Mg,n. We compute the automorphism groups of most of the Hassett spaces appearing in Kapranov’s blow-up construction. Furthermore, if g ≥ 1 we compute the automorphism groups of all Hassett spaces. In particular, we prove that if g ≥ 1 and 2g − 2 +n ≥ 3, then the automorphism groups of both Mg,A[n] and Mg,A[n] are isomorphic to a subgroup of Sn whose elements are permutations preserving the weight data in a suitable sense
Explicit log Fano structures on blow-ups of projective spaces
No full textIn this paper, we determine which blow-ups X of is ample and the pair (X,Δ) is klt. For these blow-ups, we produce explicit boundary divisors Δ making X log Fano
Birational aspects of the geometry of Varieties of Sums of Powers
No full textVarieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables. © 2013 Elsevier Ltd
Erratum: The rank of n × n matrix multiplication is at least 3 n 2 - 2 2 n 3 2 - 3 n (Linear Algebra and Its Applications (2013) 438:11 (4500-4509))
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