Green–Lazarsfeld's conjecture for generic curves of large gonality

We use Green's canonical syzygy conjecture for generic curves to prove that the Green–Lazarsfeld gonality conjecture holds for generic curves of genus g, and gonality d, if g/3<d<[g/2]+2. To cite this article: M. Aprodu, C. Voisin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

The Grothendieck conjecture for affine curves

We prove that the isomorphism class of an affine hyperbolic curve defined over a field finitely generated over Q is completely determined by its arithmetic fundamental group. We also prove a similar result for an affine curve defined over a finite field.     

Serre-Taubes duality for pseudoholomorphic curves

According to Taubes, the Gromov invariants of a symplectic four-manifold X with b₊>1 satisfy the duality Gr(α)=±Gr(κ−α), where κ is Poincaré dual to the canonical class. Extending joint work with Simon Donaldson, we interpret this result in terms of Serre duality on the fibres of a Lefschetz pencil on X, by proving an analogous symmetry for invariants counting sections of associated bundles of symmetric products. Using similar methods, we give a new proof of an existence theorem for symplectic surfaces in four-manifolds with b₊=1 and b₁=0. This reproves another theorem due to Taubes: two symplectic homology projective planes with negative canonical class and equal volume are symplectomorphic.     

The stable forking conjecture and generic structures

 We prove that for any simple theory which is constructed via Fr?issé-Hrushovski method, if the forking independence is the same as the d-independence then the stable forking property holds. Received: 22 January 2001 / Published online: 19 December 2002 This article is part of the author's D-Phil thesis, written at the University of Oxford and supported by the Ministry of Higher Education of Iran. The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for its financial support whilst working on this article. Mathematics Subject Classification (2000): 03C45 Key words or phrases: Generic structures – Fr?issé-Hrushovski method – Predimension – Simple theories – Stable theories – Stable forking conjecture     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Mond's conjecture for maps between curves

A theorem by D. Mond shows that if is finite and has has degree one onto its image (Y , 0), then the ‐codimension is less than or equal to the image Milnor number , with equality if and only if (Y , 0) is weighted homogeneous. Here we generalize this result to the case of a map germ , where (X , 0) is a plane curve singularity.     

Sheaves on abelian surfaces and strange duality

We formulate three versions of a strange duality conjecture for sections of the Theta bundles on the moduli spaces of sheaves on abelian surfaces. As supporting evidence, we check the equality of dimensions on dual moduli spaces, answering a question raised by Göttsche et al. (K-theoretic Donaldson invariants via instanton counting. arXiv:math/0611945).     

Saari's conjecture is true for generic vector fields

The simplest non-collision solutions of the -body problem are the ``relative equilibria', in which each body follows a circular orbit around the centre of mass and the shape formed by the bodies is constant. It is easy to see that the moment of inertia of such a solution is constant. In 1970, D. Saari conjectured that the converse is also true for the planar Newtonian -body problem: relative equilibria are the only constant-inertia solutions. A computer-assisted proof for the 3-body case was recently given by R. Moeckel, Trans. Amer. Math. Soc. (2005). We present a different kind of answer: proofs that several generalisations of Saari's conjecture are generically true. Our main tool is jet transversality, including a new version suitable for the study of generic potential functions.

     

Varieties of strange plane curves

Abstract

Let A be a commutative ring with identity, let X, Y be indeterminates and let F(X,Y), G(X, Y) ∈ A[X, Y] be homogeneous. Then the pair F(X, Y), G(X, Y) is said to be radical preserving with respect to A if Rad((F(x, y), G(x, y))R) = Rad((x,y)R) for each A-algebra R and each pair of elements x, y in R. It is shown that infinite sequences of pairwise radical preserving polynomials can be obtained by homogenizing cyclotomic polynomials, and that under suitable conditions on a ?-graded ring A these can be used to produce an infinite set of homogeneous prime ideals between two given homogeneous prime ideals P ? Q of A such that ht(Q/P) = 2.     

Shape-restricted inference for Lorenz curves using duality theory

We propose a new methodology for estimating curves under a convexity restriction based on Fenchel duality and wavelet approximations. In contrast to approaches where a possibly non-convex estimator is convexified at a second stage, our procedure allows us to construct directly an estimator with a convex shape. The method is applied to the estimation of the Lorenz curve. Applications to estimation of average value at risk, as well as multivariate generalisations to Lorenz surfaces are mentioned. We show asymptotic efficiency which demonstrates that the convexity is achieved at no extra cost.     

Twisted endoscopy and the generic packet conjecture

We prove a twisted analogue of the result of Rodier and Moeglin-Waldspurger on the dimension of the space of degenerate Whittaker vectors. This allows us to prove that certain twisted endoscopy forGL(n) implies the (local) generic packet conjecture for many classical groups. Partially supported by the Grants-in-Aid for Scientific Research No. 12740018, the Ministry of Education, Science, Sports and Culture, Japan.     

Tetrahedral curves via graphs and Alexander duality

A tetrahedral curve is a (usually nonreduced) curve in P³ defined by an unmixed, height two ideal generated by monomials. We characterize when these curves are arithmetically Cohen-Macaulay by associating a graph with each curve and, using results from combinatorial commutative algebra and Alexander duality, relating the structure of the complementary graph to the Cohen-Macaulay property.