A speciality theorem for curves in P<Superscript>5</Superscript>

Let be an integral projective curve. One defines the speciality index e(C) of C as the maximal integer t such that , where ω _C denotes the dualizing sheaf of . Extending a classical result of Halphen concerning the speciality of a space curve, in the present paper we prove that if is an integral degree d curve not contained in any surface of degree  < s, in any threefold of degree  < t, and in any fourfold of degree  < u, and if , then Moreover equality holds if and only if C is a complete intersection of hypersurfaces of degrees u, , and . We give also some partial results in the general case , .      

Non-free isometric immersions of Riemannian manifolds

Let (V, g) be a Riemannian manifold and let be the isometric immersion operator which, to a map , associates the induced metric on V, where denotes the Euclidean scalar product in . By Nash–Gromov implicit function theorem is infinitesimally invertible over the space of free maps. In this paper we study non-free isometric immersions . We show that the operator (where denotes the space of C ^∞- smooth quadratic forms on ) is infinitesimally invertible over a non-empty open subset of and therefore is an open map in the respective fine topologies.      

The symplectomorphism group of a blow up

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp and Diff of its one point blow up . There are three main arguments. The first shows that for any oriented M the natural map from to is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group that persist into the space of self-homotopy equivalences of . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M, a), that is, to manifolds of dimension 2n that support a class such that . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the four-torus with k-fold blow up (where k > 0) then is not generated by the groups as ranges over the set of all symplectic forms on . Partially supported by NSF grants DMS 0305939 and 0604769.     

A group-theoretical finiteness theorem

We start with the universal covering space of a closed n-manifold and with a tree of fundamental domains which zips it . Our result is that, between T and , is an intermediary object, , obtained by zipping, such that each fiber of p is finite and admits a section.      

Asymptotic analysis of solutions to parabolic systems

We study asymptotics as of solutions to a linear, parabolic system of equations with time-dependent coefficients in , where is a bounded domain. On we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function . This includes in particular situations when the coefficients may take different values on different parts of and the boundaries between them can move with t but stabilize as . The main result is an asymptotic representation of solutions for large t. A consequence is that for , the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.     

On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.      

Supremizers of inner <Emphasis Type="Italic">γ</Emphasis>-convex functions

A real-valued function f defined on a convex subset D of some normed linear space is said to be inner γ-convex w.r.t. some fixed roughness degree γ > 0 if there is a such that holds for all satisfying ||x ₀ − x ₁|| = νγ and . This kind of roughly generalized convex functions is introduced in order to get some properties similar to those of convex functions relative to their supremum. In this paper, numerous properties of their supremizers are given, i.e., of such satisfying lim . For instance, if an upper bounded and inner γ-convex function, which is defined on a convex and bounded subset D of some inner product space, has supremizers, then there exists a supremizer lying on the boundary of D relative to aff D or at a γ-extreme point of D, and if D is open relative to aff D or if dim D ≤ 2 then there is certainly a supremizer at a γ-extreme point of D. Another example is: if D is an affine set and is inner γ-convex and bounded above, then for all , and if 2 ≤ dim D < ∞ then each is a supremizer of f.      

Deforming Curves in Jacobians to Non-Jacobians I: Curves in <Emphasis Type="Italic">C</Emphasis><Superscript>(2)</Superscript>

We introduce deformation theoretic methods for determining when a curve X in a nonhyperelliptic Jacobian JC will deform with JC to a non-Jacobian. We apply these methods to a particular class of curves in the second symmetric power of C. More precisely, given a pencil of degree d on C, let X be the curve parametrizing pairs of points in divisors of (see the paper for the precise scheme-theoretical definition). We prove that if X deforms infinitesimally out of the Jacobian locus with JC then either d=4 or d=5, dim H° and C has genus 4 This material is based upon work partially supported by the National Security Agency under Grant No. MDA904-98-1-0014 and the National Science Foundation under Grant No. DMS-0071795. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Security Agency (NSA)     

Reconstruction and Subgaussian Operators in Asymptotic Geometric Analysis

We present a randomized method to approximate any vector from a set . The data one is given is the set T, vectors of and k scalar products , where are i.i.d. isotropic subgaussian random vectors in , and . We show that with high probability, any for which is close to the data vector will be a good approximation of , and that the degree of approximation is determined by a natural geometric parameter associated with the set T. We also investigate a random method to identify exactly any vector which has a relatively short support using linear subgaussian measurements as above. It turns out that our analysis, when applied to {−1, 1}-valued vectors with i.i.d. symmetric entries, yields new information on the geometry of faces of a random {−1, 1}-polytope; we show that a k- dimensional random {−1, 1}-polytope with n vertices is m-neighborly for The proofs are based on new estimates on the behavior of the empirical process when F is a subset of the L ₂ sphere. The estimates are given in terms of the γ ₂ functional with respect to the ψ ₂ metric on F, and hold both in exponential probability and in expectation. Received: November 2005, Revision: May 2006, Accepted: June 2006     

Necessary and sufficient conditions for tight equi-difference conflict-avoiding codes of weight three

A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of such that holds for any , , where . A CAC with maximum code size for given n and k is called optimal. Furthermore, an optimal CAC C is said to be tight equi-difference if holds and any codeword has the form . The concept of a CAC is motivated from applications in multiple-access communication systems. In this paper, we give a necessary and sufficient condition to construct tight equi-difference CACs of weight k = 3 and characterize the code length n’s admitting the condition through a number theoretical approach.      

Arboreal Galois representations

Let be the absolute Galois group of , and let T be the complete rooted d-ary tree, where d ≥ 2. In this article, we study “arboreal” representations of into the automorphism group of T, particularly in the case d = 2. In doing so, we propose a parallel to the well-developed and powerful theory of linear p-adic representations of . We first give some methods of constructing arboreal representations and discuss a few results of other authors concerning their size in certain special cases. We then discuss the analogy between arboreal and linear representations of . Finally, we present some new examples and conjectures, particularly relating to the question of which subgroups of Aut(T) can occur as the image of an arboreal representation of .      

An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures

We study hypersurfaces in Euclidean space whose position vector x satisfies the condition L _k x = Ax + b, where L _k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed , is a constant matrix and is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form , with . This extends a previous classification for hypersurfaces in satisfying , where is the Laplacian operator of the hypersurface, given independently by Hasanis and Vlachos [J. Austral. Math. Soc. Ser. A 53, 377–384 (1991) and Chen and Petrovic [Bull. Austral. Math. Soc. 44, 117–129 (1991)].      

Symplectic structure and reduction on the space of Riemannian metrics

The space of Riemannian metrics ${\mathfrak{Met}}MThe space of Riemannian metrics on an oriented compact manifold M of dimension n = 4k − 2 is endowed with a canonical presymplectic structure and a moment map [cf. Ferreiro Pérez and Mu?oz Masqué, Preprint (arXiv: math.DG/0507075)]. The fiber is characterized as the space of solutions to a differential equation. In dimension 2, the symplectic reduction of is analyzed and the construction presented here is compared with that introduced in Donaldson (Fields Medallists’ Lectures, 1997) and Fujiki (Sugaku Expositions 5(2):173–191, 1992). Finally, conformally flat metrics and, for n = 6, K?hler metrics of constant holomorphic sectional curvature are shown to be contained in .      

A perfect stratification of $${\mathcal M_ g}$$ for g ≤ 5

We find for g ≤ 5 a stratification of depth g − 2 of the moduli space of curves with the property that its strata are affine and the classes of their closures provide a -basis for the Chow ring of . The first property confirms a conjecture of one of us. The way we establish the second property yields new (and simpler) proofs of theorems of Faber and Izadi which, taken together, amount to the statement that in this range the Chow ring is generated by the λ-class.      

Ample vector bundles with zero loci having a bielliptic curve section of low degree

Let be an ample vector bundle of rank r ≥ 2 on a smooth complex projective variety X of dimension n such that there exists a global section of whose zero locus Z is a smooth subvariety of dimension n − r ≥ 3 of X. Let H be an ample line bundle on X such that its restriction H _Z to Z is very ample. Triplets are classified under the assumption that (Z,H _Z ) has a smooth bielliptic curve section of genus ≥ 3 with .      

<Emphasis Type="Italic">h</Emphasis>-Principles for curves and knots of constant curvature

We prove that curves of constant curvature satisfy, in the sense of Gromov, the relative -dense h-principle in the space of immersed curves in Euclidean space R ^{n ≥ 3}. In particular, in the isotopy class of any given knot f there exists a knot f͂ of constant curvature which is -close to f. More importantly, we show that if f is , then the curvature of f͂ may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that f͂ be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull. The author’s research was supported in part by NSF CAREER award DMS-0332333.     

The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity

In this paper we establish a general weighted L ^q -theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L ^q -estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L ^p -regularity of in the respective weighted L ^q -spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L ^p -regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L _p -regularity. Preprint (1999)] with the fact that for L ^q -spaces -boundedness is implied by weighted estimates.     

Generalized hexagons and Singer geometries

In this paper, we consider a set of lines of with the properties that (1) every plane contains 0, 1 or q + 1 elements of , (2) every solid contains no more than q ² + q + 1 and no less than q + 1 elements of , and (3) every point of is on q + 1 members of , and we show that, whenever (4) q ≠ 2 (respectively, q = 2) and the lines of through some point are contained in a solid (respectively, a plane), then is necessarily the set of lines of a regularly embedded split Cayley generalized hexagon in , with q even. We present examples of such sets not satisfying (4) based on a Singer cycle in , for all q.      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

Relaxation of the curve shortening flow via the parabolic Ginzburg -Landau equation

In this paper we study how to find solutions to the parabolic Ginzburg–Landau equation that as have as interface a given curve that evolves under curve shortening flow. Moreover, for compact embedded curves we find a uniform profile for the solution up the extinction time of the curve. We show that after the extinction time the solution converges uniformly to a constant.