22,607 research outputs found

    Point-curve incidences in the complex plane

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    We prove an incidence theorem for points and curves in the complex plane. Given a set of mm points in R2{\mathbb R}^2 and a set of nn curves with kk degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is O(mk2k−1n2k−22k−1+m+n)O\big(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n\big). We establish the slightly weaker bound Oε(mk2k−1+εn2k−22k−1+m+n)O_\varepsilon\big(m^{\frac{k}{2k-1}+\varepsilon}n^{\frac{2k-2}{2k-1}}+m+n\big) on the number of incidences between mm points and nn (complex) algebraic curves in C2{\mathbb C}^2 with kk 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 C{\mathbb C}.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

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    We construct an exceptional collection Υ\Upsilon of maximal possible length 6 on any of the Burniat surfaces with KX2=6K_X^2=6, a 4-dimensional family of surfaces of general type with pg=q=0p_g=q=0. 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 A\mathcal A of Υ\Upsilon 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

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    We show how to find explicit expressions for rigid string instantons for general 4-manifold MM. It appears that they are pseudo-holomorphic curves in the twistor space of MM. We present explicit formulae for M=R4,S4M=R^4, S^4. 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

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    In this paper we try to find examples of integrable natural Hamiltonian systems on the sphere S2S^2 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

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    We show that the number of unit-area triangles determined by a set SS of nn points in the plane is O(n20/9)O(n^{20/9}), improving the earlier bound O(n9/4)O(n^{9/4}) 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 SS consists of points on three lines, the number of unit-area triangles that SS spans can be Ω(n2)\Omega(n^2), for any triple of lines (it is always O(n2)O(n^2) in this case). (ii) We show that if SS is a {\em convex grid} of the form A×BA\times B, where AA, BB are {\em convex} sets of n1/2n^{1/2} real numbers each (i.e., the sequences of differences of consecutive elements of AA and of BB are both strictly increasing), then SS determines O(n31/14)O(n^{31/14}) unit-area triangles
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