131 research outputs found
Algebraic entropy and the space of initial values for discrete dynamical systems
A method to calculate the algebraic entropy of a mapping which can be lifted
to an isomorphism of a suitable rational surfaces (the space of initial values)
are presented. It is shown that the degree of the th iterate of such a
mapping is given by its action on the Picard group of the space of initial
values. It is also shown that the degree of the th iterate of every
Painlev\'e equation in sakai's list is at most and therefore its
algebraic entropy is zero.Comment: 10 pages, pLatex fil
Hori--Vafa mirror models for complete intersections in weighted projective spaces and weak Landau--Ginzburg models
We prove that Hori--Vafa mirror models for smooth Fano complete intersections
in weighted projective spaces admit an interpretation as Laurent polynomials.Comment: 5 pages; several minor changes has been mad
The rationality of the moduli spaces of Coble surfaces and of nodal Enriques surfaces
We prove the rationality of the coarse moduli spaces of Coble surfaces and of
nodal Enriques surfaces over the field of complex numbers.Comment: 15 page
The dissimilarity map and representation theory of
We give another proof that -dissimilarity vectors of weighted trees are
points on the tropical Grassmanian, as conjectured by Cools, and proved by
Giraldo in response to a question of Sturmfels and Pachter. We accomplish this
by relating -dissimilarity vectors to the representation theory of Comment: 11 pages, 8 figure
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
Shimura curve computations via K3 surfaces of Neron-Severi rank at least 19
It is known that K3 surfaces S whose Picard number rho (= rank of the
Neron-Severi group of S) is at least 19 are parametrized by modular curves X,
and these modular curves X include various Shimura modular curves associated
with congruence subgroups of quaternion algebras over Q. In a family of such K3
surfaces, a surface has rho=20 if and only if it corresponds to a CM point on
X. We use this to compute equations for Shimura curves, natural maps between
them, and CM coordinates well beyond what could be done by working with the
curves directly as we did in ``Shimura Curve Computations'' (1998) =
Comment: 16 pages (1 figure drawn with the LaTeX picture environment); To
appear in the proceedings of ANTS-VIII, Banff, May 200
K3-fibered Calabi-Yau threefolds I, the twist map
A construction of Calabi-Yaus as quotients of products of lower-dimensional
spaces in the context of weighted hypersurfaces is discussed, including
desingularisation. The construction leads to Calabi-Yaus which have a fiber
structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus
are of some interest in connection with Type II -heterotic string dualities in
dimension 4. A section at the end of the paper summarises this for the
non-expert mathematician.Comment: 31 pages LaTeX, 11pt, 2 figures. To appear in International Journal
of Mathematics. On the web at
http://personal-homepages.mis.mpg.de/bhunt/preprints.html , #
Apolarity, Hessian and Macaulay polynomials
A result by Macaulay states that an Artinian graded Gorenstein ring R of
socle dimension one and socle degree b can be realized as the apolar ring of a
homogeneous polynomial f of degree b. If R is the Jacobian ring of a smooth
hypersurface g=0, then b is just equal to the degree of the Hessian polynomial
of g. In this paper we investigate the relationship between f and the Hessian
polynomial of g.Comment: 12 pages. Improved exposition, minor correction
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
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