The automorphisms group of M‾0,n\overline{M}_{0,n}M0,n (original) (raw)

2013, Journal of the European Mathematical Society

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Abstract

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This research explores the automorphism group of the moduli space M_0,n of smooth n-pointed curves of genus 0. Leveraging Kapranov's description, it establishes that the only bijective biregular automorphisms are permutations of the markings on the curves, specifically confirming that Aut(M_0,n) is isomorphic to the symmetric group S_n for n ≥ 5. The findings provide a comprehensive classification and understanding of the structure of automorphisms in this moduli space.

Forgetful linear systems on the projective space and rational normal curves over _0,2n^GIT

2009

Let _0,n the moduli space of n-pointed rational curves. The aim of this note is to give a new, geometric construction of _0,2n^GIT, the GIT compacification of _0,2n, in terms of linear systems on ^2n-2 that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between _0,2n+1 and _0,2n. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactification _0,2n to _0,2n^GIT and of the canonical forgetful maps. As a side result we also find a linear system on _0,2n whose associated map is the contraction map c_2n.

Forgetful Linear Systems on the Projective Space and Rational Normal Curves over M

Let M 0,n the moduli space of n-pointed rational curves. The aim of this note is to give a new, geometric construction of M GIT 0,2n , the GIT com-pacification of M 0,2n , in terms of linear systems on P 2n−2 that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between M 0,2n+1 and M 0,2n. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactification M 0,2n to M GIT 0,2n and of the canonical forgetful maps.

Birational geometry of moduli spaces of configurations of points on the line

arXiv: Algebraic Geometry, 2017

In this paper we study the geometry of GIT configurations of nnn ordered points on mathbbP1\mathbb{P}^1mathbbP1 both from the the birational and the biregular viewpoint. In particular, we prove that any extremal ray of the Mori cone of effective curves of the quotient (mathbbP1)n//PGL(2)(\mathbb{P}^1)^n//PGL(2)(mathbbP1)n//PGL(2), taken with the symmetric polarization, is generated by a one dimensional boundary stratum of the moduli space. Furthermore, we develop some technical machinery that we use to compute the canonical divisor and the Hilbert polynomial of (mathbbP1)n//PGL(2)(\mathbb{P}^1)^n//PGL(2)(mathbbP1)n//PGL(2) in its natural embedding, and its group of automorphisms.

Automorphisms in Birational and Affine Geometry

Springer Proceedings in Mathematics & Statistics, 2014

The configuration space C n (X) of an algebraic curve X is the algebraic variety consisting of all n-point subsets Q ⊂ X. We describe the automorphisms of C n (C), deduce that the (infinite dimensional) group Aut C n (C) is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of C n (C) are also computed. We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level. This is an extended version of our paper [39]. We strengthened the results concerning the automorphism groups of cylinders over rigid bases, replacing the rigidity assumption by the weaker assumption of tightness. We also added alternative proofs of two auxiliary results cited in [39] and due to Zinde and to the first author. This allowed us to provide the optimal dimension bounds in our theorems.

Automorphisms of rational double points and moduli spaces of surfaces of general type

Compositio Mathematica

L'accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Configuration Spaces of the Affine Line and their Automorphism Groups

Springer Proceedings in Mathematics & Statistics, 2014

On Plane Cremona Transformations of Fixed Degree

Journal of Geometric Analysis, 2013

The birational geometry of algebraic varieties is governed by the group of birational self-maps. It is in general very difficult to determine this group for an arbitrary variety and as a matter of facts only few examples are completely understood. The special case of the projective plane attracted lots of attention since the XIX th -century. The pioneering work of Cremona and then the classical geometers of the Italian and German school were able to give partial descriptions of it but it was only after Noether and Castelnuovo that generators of the group were described. The Noether-Castelnuovo Theorem, see for instance , states that the group of birational self-maps of P 2 , usually called the plane Cremona group and denoted by Cr , is generated by linear automorphisms of P 2 and a single birational non biregular map, the so-called elementary quadratic transformation σ : P 2 P 2 defined by σ([x : y : z]) = [yz : xz : xy]. Even if the generators of Cr(2) have been known for a century now, many other properties of this group are still mysterious. Only after decades, see , a complete set of relations has been described, and more recently the non simplicity of Cr(2) has been showed, , and a good understanding of its finite subgroups has been achieved, see [17] and [5]. This brief and fairly incomplete list is only meant to stress the difficulties and the large unknown parts in the study of Cr(2), for a more complete picture the interested reader should refer to and . Amid all its subgroups the one associated to polynomial automorphisms of the plane, Aut(C 2 ), attracted even more attention than Cr(2) itself, . The generators of Aut(C 2 ) are known since 1942, , and later on, [32], has been proved that Aut(C 2 ) is the amalgamated product of two of its subgroups, more precisely of the affine and elementary ones. Nevertheless this group is not less mysterious and challenging than the entire Cremona group. Jung's description yields a natural decomposition

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References (14)

M. Bolognesi Forgetful linear systems on the projective space and rational normal curves over M GIT 0,2n ArXiv:alg-geom0909.0151.
A. Bruno, M. Mella Fiber type morphisms of M 0,n in preparation (2010).
A. Castravet, J. Tevelev Exceptional loci on M 0,n and hypergraph curves ArXiv:alg- geom0809.1699.
C. J. Earle, I. Kra On isometries between Teichmüller spaces Duke Math. J. V.41, No. 3 (1974), 583-591.
G. Farkas, A. Gibney The Mori cones of moduli spaces of pointed curves of small genus. Trans. Amer. Math. Soc. 355 (2003), 1183-1199.
A. Gibney, The Fibrations of the Moduli Space of Stable n-Pointed Curves of Genus g, M(sub g,n), Phd thesis University of Texas at Austin (2000)
A. Gibney, S. Keel, I. Morrison Towards the ample cone of M g,n Journal of Amer. Math. Soc., 2 (2001), 273-294.
M. Kapranov Veronese curves and Grothendieck-Knudsen moduli spaces M 0,n . Jour. Alg. Geom., 2 (1993), 239-262.
S. Keel, J. M c Kernan Contractible extremal rays of M 0,n . ArXiv:alg-geom9607009.
M. Korkmaz Automorphisms of complexes of curves on punctured spheres and on punc- tured tori Topology and its Applications Volume 95, Issue 2 (1999) 85-111.
P. Larsen Fulton's conjecture for M 0,7 . ArXiv:alg-geom0912310.
A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of flat connec- tions, Michigan Math. J. Volume 48, Issue 1 (2000), 443-472 [M c T] J. M c Kernan, J. Tevelev private communication [Mo] I. Morrison Mori theory of moduli spaces of stable curves. Projective Press, New York 2007.
H.L. Royden Automorphisms and isometries of Teichmüller spaces in Advances in the theory of Riemann surfaces Ed. by L. V. Ahlfors L. Bers H. M. Farkas R. C. Gunning I. Kra H. E. Rauch Annals of Math. Studies No.66 (1971), 369-383.
Dipartimento di Matematica, Università Roma Tre, L.go S.L.Murialdo, 1,

On the Automorphisms of Moduli Spaces of Curves

Springer Proceedings in Mathematics & Statistics, 2014

In the last years the biregular automorphisms of the Deligne-Mumford's and Hassett's compactifications of the moduli space of n-pointed genus g smooth curves have been extensively studied by A. Bruno and the authors. In this paper we give a survey of these recent results and extend our techniques to some moduli spaces appearing as intermediate steps of the Kapranov's and Keel's realizations of M 0,n, and to the degenerations of Hassett's spaces obtained by allowing zero weights.

FORGETFUL LINEAR SYSTEMS ON THE PROJECTIVE SPACE AND RATIONAL NORMAL CURVES OVER M GIT 0,2n

2009

Let M0,n the moduli space of n-pointed rational curves. The aim of this note is to give a new, geometric construction of MGIT 0,2n, the GIT compacification of M0,2n, in terms of linear systems on P2n−2 that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between M0,2n+1 and M0,2n. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactification M0,2n to MGIT 0,2n and of the canonical forgetful maps.

The automorphisms group of M_0,n

2010

In this paper we study fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov's description we are able to prove that the only such morphisms are forgetful maps. This allow us to show that the Automorphism group of M_0,n is the permutation group on n elements as soon as n≥ 5.

Forgetful linear systems on the projective space and rational normal curves over $\cM_{0,2n}^{GIT

2009

Let M 0,n the moduli space of n-pointed rational curves. The aim of this note is to give a new geometric construction of M GIT 0,2n , the GIT compactification of M 0,2n , in terms of linear systems on P 2n−2 that contract all the rational normal curves passing by the points of a projective base. These linear systems are somehow a projective analogue of the forgetful maps between the Mumford-Knudsen compactifications M 0,2n+1 and M 0,2n , but on the other hand they contract some components of the boundary, yielding then a rational map onto M GIT 0,2n . The construction is performed via a study of the socalled contraction maps from M 0,2n to M GIT 0,2n and of the canonical forgetful maps. As a side result we also find a linear system on M 0,2n whose associated map is the contraction map c 2n .

A connection between birational automorphisms of the plane and linear systems of curves

Journal of Computational and Applied Mathematics, 2015

In this paper, we prove that there exits a one-to-one correspondence between birational automorphisms of the plane and pairs of pencils of curves intersecting in a unique point. As a consequence, we show how to construct birational automorphisms of the plane of a certain degree d (fixed in advance) from some curves generating two linear systems of curves of degrees d and d, where d = d -2 for d > 2, and d = 1 otherwise. In addition, we also get the inverse of the birational automorphism constructed, and we show that its degree is obtained from the degree of the linear system of curves. As a special case, we show how these results can be stated to polynomial birational automorphisms of the plane.

Topological aspects of the dynamical moduli space of rational maps. (arXiv:1908.10792v1 [math.AT])

arXiv (Cornell University), 2019

We investigate the topology of the space of Möbius conjugacy classes of degree d rational maps on the Riemann sphere. We show that it is rationally acyclic and we compute its fundamental group. As a byproduct, we also obtain the ranks of some higher homotopy groups of the parameter space of degree d rational maps allowing us to extend the previously known range. Contents 1. Introduction 2. Background on the spaces Rat d and M d 2.1. The parameter space Rat d 2.2. The moduli space M d 3. The homology of M d 3.1. The rational homology of M d 3.2. Application to marked moduli spaces of rational maps 13 3.3. Variants M pre d and M post d of the moduli space M d 15 3.4. The cohomology of M d with finite coefficients 17 4. The fundamental group of M d 22 5. The higher homotopy groups of Rat d 27 References 32

Automorphism loci for degree 3 and degree 4 endomorphisms of the projective line

arXiv: Dynamical Systems, 2020

Let fff be an endomorphism of the projective line. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group. The group of automorphisms, or stabilizer group, of a given fff for this action is known to be a finite group. We determine explicit families that parameterize all endomorphisms defined over barmathbbQ\bar{\mathbb{Q}}barmathbbQ of degree 333 and degree 444 that have a nontrivial automorphism, the \textit{automorphism locus} of the moduli space of dynamical systems. We analyze the geometry of these loci in the appropriate moduli space of dynamical systems. Further, for each family of maps, we study the possible structures of mathbbQ\mathbb{Q}mathbbQ-rational preperiodic points which occur under specialization.

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Birational geometry of moduli spaces of configurations of points on the line

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The automorphism group ofM0,ntropandM‾0,ntrop

Journal of Combinatorial Theory, Series A, 2018

In this paper we show that the automorphism groups of M trop 0,n and M trop 0,n are isomorphic to the permutation group Sn for n ≥ 5, while the automorphism groups of M trop 0,4 and M trop 0,4 are isomorphic to the permutation group S 3 .

The automorphisms group of M‾0,n\overline{M}_{0,n}M0,n​ (original) (raw)

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The automorphisms group of M‾0,n\overline{M}_{0,n}M0,n (original) (raw)