110 research outputs found
Normal subgroups in the Cremona group (long version)
Let k be an algebraically closed field. We show that the Cremona group of all
birational transformations of the projective plane P^2 over k is not a simple
group. The strategy makes use of hyperbolic geometry, geometric group theory,
and algebraic geometry to produce elements in the Cremona group that generate
non trivial normal subgroups.Comment: With an appendix by Yves de Cornulier. Numerous but minors
corrections were made, regarding proofs, references and terminology. This
long version contains detailled proofs of several technical lemmas about
hyperbolic space
Curves on Heisenberg invariant quartic surfaces in projective 3-space
This paper is about the family of smooth quartic surfaces that are invariant under the Heisenberg group . For a
very general such surface , we show that the Picard number of is 16 and
determine its Picard group. It turns out that the general Heisenberg invariant
quartic contains 320 smooth conics and that in the very general case, this
collection of conics generates the Picard group.Comment: Updated references, corrected typo
Dark energy domination in the Virgocentric flow
The standard \LambdaCDM cosmological model implies that all celestial bodies
are embedded in a perfectly uniform dark energy background, represented by
Einstein's cosmological constant, and experience its repulsive antigravity
action. Can dark energy have strong dynamical effects on small cosmic scales as
well as globally? Continuing our efforts to clarify this question, we focus now
on the Virgo Cluster and the flow of expansion around it. We interpret the
Hubble diagram, from a new database of velocities and distances of galaxies in
the cluster and its environment, using a nonlinear analytical model which
incorporates the antigravity force in terms of Newtonian mechanics. The key
parameter is the zero-gravity radius, the distance at which gravity and
antigravity are in balance. Our conclusions are: 1. The interplay between the
gravity of the cluster and the antigravity of the dark energy background
determines the kinematical structure of the system and controls its evolution.
2. The gravity dominates the quasi-stationary bound cluster, while the
antigravity controls the Virgocentric flow, bringing order and regularity to
the flow, which reaches linearity and the global Hubble rate at distances \ga
15 Mpc. 3. The cluster and the flow form a system similar to the Local Group
and its outflow. In the velocity-distance diagram, the cluster-flow structure
reproduces the group-flow structure with a scaling factor of about 10; the
zero-gravity radius for the cluster system is also 10 times larger. The phase
and dynamical similarity of the systems on the scales of 1-30 Mpc suggests that
a two-component pattern may be universal for groups and clusters: a
quasi-stationary bound central component and an expanding outflow around it,
due to the nonlinear gravity-antigravity interplay with the dark energy
dominating in the flow component.Comment: 7 pages, 2 figures, Astronomy and Astrophysics (accepted
Stratifying quotient stacks and moduli stacks
Recent results in geometric invariant theory (GIT) for non-reductive linear
algebraic group actions allow us to stratify quotient stacks of the form [X/H],
where X is a projective scheme and H is a linear algebraic group with
internally graded unipotent radical acting linearly on X, in such a way that
each stratum [S/H] has a geometric quotient S/H. This leads to stratifications
of moduli stacks (for example, sheaves over a projective scheme) such that each
stratum has a coarse moduli space.Comment: 25 pages, submitted to the Proceedings of the Abel Symposium 201
Del Pezzo surfaces with many symmetries
We classify smooth del Pezzo surfaces whose alpha-invariant of Tian is bigger
than one.Comment: 23 page
Three embeddings of the Klein simple group into the Cremona group of rank three
We study the action of the Klein simple group G consisting of 168 elements on
two rational threefolds: the three-dimensional projective space and a smooth
Fano threefold X of anticanonical degree 22 and index 1. We show that the
Cremona group of rank three has at least three non-conjugate subgroups
isomorphic to G. As a by-product, we prove that X admits a Kahler-Einstein
metric, and we construct a smooth polarized K3 surface of degree 22 with an
action of the group G.Comment: 43 page
Polar Cremona Transformations and Monodromy of Polynomials
Consider the gradient map associated to any non-constant homogeneous
polynomial f\in \C[x_0,...,x_n] of degree , defined by \phi_f=grad(f):
D(f)\to \CP^n, (x_0:...:x_n)\to (f_0(x):...:f_n(x)) where D(f)=\{x\in \CP^n;
f(x)\neq 0\} is the principal open set associated to and
. This map corresponds to polar Cremona
transformations. In Proposition \ref{p1} we give a new lower bound for the
degree of under the assumption that the projective hypersurface
has only isolated singularities. When , Theorem \ref{t4}
yields very strong conditions on the singularities of .Comment: 8 page
From SICs and MUBs to Eddington
This is a survey of some very old knowledge about Mutually Unbiased Bases
(MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions
the former are closely tied to an elliptic normal curve symmetric under the
Heisenberg group, while the latter are believed to be orbits under the
Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are
understandable in terms of elliptic curves, but a general statement escapes us.
The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber
Local dark energy: HST evidence from the expansion flow around Cen A/M83 galaxy group
A structure with a massive group in its center and a cool expansion outflow
outside is studied around the Cen A galaxy with the use of the Hubble Space
Telescope observations. It is demonstrated that the dynamics of the flow is
dominated by the antigravity of the dark energy background. The density of dark
energy in the cell is estimated to be near the global cosmological density.
This agrees with our previous result from the neighborhood of the Local group.
A notion of the ``Hubble cell'' is introduced as a building block of the local
structure of the universe
Matrix factorisations and D-branes on K3
D-branes on K3 are analysed from three different points of view. For
deformations of hypersurfaces in weighted projected space we use geometrical
methods as well as matrix factorisation techniques. Furthermore, we study the
D-branes on the T^4/\Z_4 orbifold line in conformal field theory. The behaviour
of the D-branes under deformations of the bulk theory are studied in detail,
and good agreement between the different descriptions is found.Comment: 35 pages, no figure
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