Spaces of rational loops on a real projective space

We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps . As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.

     

Number of singularities of a foliation on

Let be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let be the dimension of the singular locus of . We give an upper bound of the number of singularities of dimension , counted with multiplicity and degree, that may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.

     

Rational surfaces with

The main but not all of the results in this paper concern rational surfaces for which the self-intersection of the anticanonical class is positive. In particular, it is shown that no superabundant numerically effective divisor classes occur on any smooth rational projective surface with . As an application, it follows that any 8 or fewer (possibly infinitely near) points in the projective plane are in good position. This is not true for 9 points, and a characterization of the good position locus in this case is also given. Moreover, these results are put into the context of conjectures for generic blowings up of . All results are proven over an algebraically closed field of arbitrary characteristic.

     

Rationality of the quotient of ℙ2 by finite group of automorphisms over arbitrary field of characteristic zero

Let $\Bbbk$ be a field of characteristic zero and G be a finite group of automorphisms of projective plane over $\Bbbk$ . Castelnuovo’s criterion implies that the quotient of projective plane by G is rational if the field $\Bbbk$ is algebraically closed. In this paper we prove that ${{\mathbb{P}_\Bbbk ^2 } \mathord{\left/ {\vphantom {{\mathbb{P}_\Bbbk ^2 } G}} \right. \kern-0em} G}$ is rational for an arbitrary field $\Bbbk$ of characteristic zero.     

Non-existence of a curve over of genus 5 with 14 rational points

We show that an absolutely irreducible, smooth, projective curve of genus over with rational points cannot exist.

     

Gorenstein derived functors

Over any associative ring it is standard to derive using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product using Gorenstein flat modules.

     

Homological algebra for the representation Green functor for abelian groups

In this paper we compute some derived functors of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic -group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor .

When the group is , we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired functors.

     

On conformally Kähler, Einstein manifolds

We prove that any compact complex surface with admits an Einstein metric which is conformally related to a Kähler metric. The key new ingredient is the existence of such a metric on the blow-up of the complex projective plane at two distinct points.

     

Quantum cohomology of projective bundles over

In this paper we study the quantum cohomology ring of certain projective bundles over the complex projective space . Using excessive intersection theory, we compute the leading coefficients in the relations among the generators of the quantum cohomology ring structure. In particular, Batyrev's conjectural formula for quantum cohomology of projective bundles associated to direct sum of line bundles over is partially verified. Moreover, relations between the quantum cohomology ring structure and Mori's theory of extremal rays are observed. The results could shed some light on the quantum cohomology for general projective bundles.

     

-saddle method and Beukers' integral

We give good non-quadraticity measures for the values of logarithm at specific rational points by modifying Beukers' double integral. The two-dimensional version of the saddle method, which we call -saddle method, is applied.

     

Condensations of projective sets onto compacta

For a coanalytic-complete or -complete subspace of a Polish space we prove that there exists a continuous bijection of onto the Hilbert cube . This extends results of Pytkeev. As an application of our main theorem we give an answer to some questions of Arkhangelskii and Christensen.

Under the assumption of Projective Determinacy we also give some generalizations of these results to higher projective classes.

     

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on

The local Phragmén-Lindelöf condition for analytic subvarieties of  at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in  . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on  .

     

Sieving for rational points on hyperelliptic curves

We give a new and efficient method of sieving for rational points on hyperelliptic curves. This method is often successful in proving that a given hyperelliptic curve, suspected to have no rational points, does in fact have no rational points; we have often found this to be the case even when our curve has points over all localizations . We illustrate the practicality of the method with some examples of hyperelliptic curves of genus .

     

Degree of strata of singular cubic surfaces

We determine the degree of some strata of singular cubic surfaces in the projective space . These strata are subvarieties of the parametrizing all cubic surfaces in . It is known what their dimension is and that they are irreducible. In 1986, D. F. Coray and I. Vainsencher computed the degree of the 4 strata consisting on cubic surfaces with a double line. To work out the case of isolated singularities we relate the problem with (stationary) multiple-point theory.

     

Galois groups via Atkin-Lehner twists

Using Serre's proposed complement to Shih's Theorem, we obtain as a Galois group over for at least new primes . Assuming that rational elliptic curves with odd analytic rank have positive rank, we obtain Galois realizations for of the primes that were not covered by previous results; it would also suffice to assume a certain (plausible, and perhaps tractable) conjecture concerning class numbers of quadratic fields. The key issue is to understand rational points on Atkin-Lehner twists of . In an appendix, we explore the existence of local points on these curves.

     

Uniqueness of exceptional singular quartics

We prove that given a general collection of 14 points of ( an infinite field) there is a unique quartic hypersurface that is singular on .

This completes the solution to the open problem of the dimension of a linear system of hypersurfaces of that are singular on a collection of general points.

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.

     

Intersection theory on and elliptic Gromov-Witten invariants

We find a new relation among codimension algebraic cycles in the moduli space , and use this to calculate the elliptic Gromov-Witten invariants of projective spaces and .

     

Characterizations of regular almost periodicity in compact minimal abelian flows

Regular almost periodicity in compact minimal abelian flows was characterized for the case of discrete acting group by W. Gottschalk and G. Hedlund and for the case of -dimensional phase space by W. Gottschalk a few decades ago. In 1995 J. Egawa gave characterizations for the case when the acting group is . We extend Egawa's results to the case of an arbitrary abelian acting group and a not necessarily metrizable phase space. We then show how our statements imply previously known characterizations in each of the three special cases and give various other applications (characterization of regularly almost periodic functions on arbitrary abelian topological groups, classification of uniformly regularly almost periodic compact minimal - and -flows, conditions equivalent with uniform regular almost periodicity, etc.).