22,607 research outputs found
Point-curve incidences in the complex plane
We prove an incidence theorem for points and curves in the complex plane.
Given a set of points in and a set of curves with
degrees of freedom, Pach and Sharir proved that the number of point-curve
incidences is . We
establish the slightly weaker bound
on the number of incidences between points and (complex) algebraic
curves in with degrees of freedom. We combine tools from
algebraic geometry and differential geometry to prove a key technical lemma
that controls the number of complex curves that can be contained inside a real
hypersurface. This lemma may be of independent interest to other researchers
proving incidence theorems over .Comment: The proof was significantly simplified, and now relies on the
Picard-Lindelof theorem, rather than on foliation
Derived categories of Burniat surfaces and exceptional collections
We construct an exceptional collection of maximal possible length
6 on any of the Burniat surfaces with , a 4-dimensional family of
surfaces of general type with . We also calculate the DG algebra of
endomorphisms of this collection and show that the subcategory generated by
this collection is the same for all Burniat surfaces.
The semiorthogonal complement of is an "almost
phantom" category: it has trivial Hochschild homology, and K_0(\mathcal
A)=\bZ_2^6.Comment: 15 pages, 1 figure; further remarks expande
Rigid string instantons are pseudo-holomorphic curves
We show how to find explicit expressions for rigid string instantons for
general 4-manifold . It appears that they are pseudo-holomorphic curves in
the twistor space of . We present explicit formulae for . We
discuss their properties and speculate on relations to topology of 4-manifolds
and the theory of Yang-Mills fields.Comment: 18 pages,Late
Searching for integrable Hamiltonian systems with Platonic symmetries
In this paper we try to find examples of integrable natural Hamiltonian
systems on the sphere with the symmetries of each Platonic polyhedra.
Although some of these systems are known, their expression is extremely
complicated; we try here to find the simplest possible expressions for this
kind of dynamical systems. Even in the simplest cases it is not easy to prove
their integrability by direct computation of the first integrals, therefore, we
make use of numerical methods to provide evidences of integrability; namely, by
analyzing their Poincar\'e sections (surface sections). In this way we find
three systems with platonic symmetries, one for each class of equivalent
Platonic polyhedra: tetrahedral, exahedral-octahedral,
dodecahedral-icosahedral, showing evidences of integrability. The proof of
integrability and the construction of the first integrals are left for further
works. As an outline of the possible developments if the integrability of these
systems will be proved, we show how to build from them new integrable systems
in dimension three and, from these, superintegrable systems in dimension four
corresponding to superintegrable interactions among four points on a line, in
analogy with the systems with dihedral symmetry treated in a previous article.
A common feature of these possibly integrable systems is, besides to the rich
symmetry group on the configuration manifold, the partition of the latter into
dynamically separated regions showing a simple structure of the potential in
their interior. This observation allows to conjecture integrability for a class
of Hamiltonian systems in the Euclidean spaces.Comment: 22 pages; 4 figure
The number of unit-area triangles in the plane: Theme and variations
We show that the number of unit-area triangles determined by a set of
points in the plane is , improving the earlier bound
of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. We also consider two
special cases of this problem: (i) We show, using a somewhat subtle
construction, that if consists of points on three lines, the number of
unit-area triangles that spans can be , for any triple of
lines (it is always in this case). (ii) We show that if is a {\em
convex grid} of the form , where , are {\em convex} sets of
real numbers each (i.e., the sequences of differences of consecutive
elements of and of are both strictly increasing), then determines
unit-area triangles
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