989 research outputs found
Algorithms for Del Pezzo Surfaces of Degree 5 (Construction, Parametrization)
It is well known that every Del Pezzo surface of degree 5 defined over k is
parametrizable over k. In this paper we give an efficient construction for
parametrizing, as well as algorithms for constructing examples in every
isomorphism class and for deciding equivalence.Comment: 15 page
Counting points on curves over families in polynomial time
This note concerns the theoretical algorithmic problem of counting rational
points on curves over finite fields. It explicates how the algorithmic scheme
introduced by Schoof and generalized by the author yields an algorithm whose
running time is uniformly polynomial time for curves in families.Comment: 7 page
Pre-images of quadratic dynamical systems
For a quadratic endomorphism of the affine line defined over the rationals we
consider the problem of bounding the number of rational points that eventually
land at a given constant after iteration, called pre-images of the constant. In
the article "Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,"
it was shown that the number of rational pre-images is bounded as one varies
the morphism in a certain one-dimensional family. Explicit values of the
constant for pre-images of zero and -1 defined over the rational numbers were
addressed in subsequent articles. This article addresses an explicit bound for
any algebraic image constant and provides insight into the geometry of the
"pre-image surfaces."Comment: to appear in Involve; 16page
The enumeration of simultaneous higher-order contacts between plane curves
Using the Semple bundle construction, we derive an intersection-theoretic
formula for the number of simultaneous contacts of specified orders between
members of a generic family of degree plane curves and finitely many fixed
curves. The contacts counted by the formula occur at nonsingular points of both
the members of the family and the fixed curves.Comment: 32 pages, AmS-TeX v2.1 (Revised statement and proof of one lemma;
other minor changes.
Orbit Parametrizations for K3 Surfaces
We study moduli spaces of lattice-polarized K3 surfaces in terms of orbits of
representations of algebraic groups. In particular, over an algebraically
closed field of characteristic 0, we show that in many cases, the nondegenerate
orbits of a representation are in bijection with K3 surfaces (up to suitable
equivalence) whose N\'eron-Severi lattice contains a given lattice. An
immediate consequence is that the corresponding moduli spaces of these
lattice-polarized K3 surfaces are all unirational. Our constructions also
produce many fixed-point-free automorphisms of positive entropy on K3 surfaces
in various families associated to these representations, giving a natural
extension of recent work of Oguiso.Comment: 83 pages; to appear in Forum of Mathematics, Sigm
Darboux cyclides and webs from circles
Motivated by potential applications in architecture, we study Darboux
cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin
cyclides and quadrics, and they carry up to six real families of circles.
Revisiting the classical approach to these surfaces based on the spherical
model of 3D Moebius geometry, we provide computational tools for the
identification of circle families on a given cyclide and for the direct design
of those. In particular, we show that certain triples of circle families may be
arranged as so-called hexagonal webs, and we provide a complete classification
of all possible hexagonal webs of circles on Darboux cyclides.Comment: 34 pages, 20 figure
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