The density of rational points on a certain singular cubic surface

We show that the number of nontrivial rational points of height at most B, which lie on the cubic surface x₁x₂x₃=x₄²(x₁+x₂+x₃), has order of magnitude B⁶(logB). This agrees with Manin's conjecture.     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003     

On the number of rational points on a strictly convex curve

Let γ be a bounded convex curve on the plane. Then #(γ ∩ (?/n)²) = o(n ^2/3). This strengthens the classical result due to Jarník [J] (the upper bound cn ^2/3) and disproves the conjecture on the existence of a so-called universal Jarník curve.     

Gap orders of rational functions on plane curves with few singular points

We prove that certain integers n cannot occur as degrees of linear series without base points on the normalization of a plane curve whose only singularities are a “small” number of nodes and ordinary cusps. As a consequence we compute the gonality of such a curve. Work done with financial support of M.U.R.S.T. while the authors were members of C.N.R.     

The canonical model of a singular curve

We give refined statements and modern proofs of Rosenlicht’s results about the canonical model C′ of an arbitrary complete integral curve C. Notably, we prove that C and C′ are birationally equivalent if and only if C is nonhyperelliptic, and that, if C is nonhyperelliptic, then C′ is equal to the blowup of C with respect to the canonical sheaf ω. We also prove some new results: we determine just when C′ is rational normal, arithmetically normal, projectively normal, and linearly normal.      

Factorization of a rational matrix: The singular case

We study the problem of minimal factorization of an arbitrary rational matrix R(), i. e. where R() is not necessarily square or invertible. Following the definition of minimality used here, we show that the problem can be solved via a generalized eigenvalue problem which will be singular when R() is singular. The concept of invariant subspace, which has been used in the solution of the minimal factorization problem for regular matrices, is now replaced by a reducing subspace, a recently introduced concept which is a logical extension of invariant and deflating subspaces to the singular pencil case.     

Separatrices and singular points

A new definition of separatrix is introduced and the theory developed is applied to the study of the topological structure of singular points. Each component of the complement of the separatrix set in a neighborhood of an isolated singular point is shown to satisfy one of seven types of behavior.     

Curves with a prescribed number of rational points

We show that for any finite field Fq, any N?0 and all sufficiently large integers g there exist curves over Fq of genus g having exactly N rational points.     

Isomonodromic confluence of singular points

We consider the problem of confluence of singular points under isomonodromic deformations of linear systems. We prove that a system with irregular singular points is a result of isomonodromic confluence of singular points with minimal Poincaré ranks, i.e., of singular points whose Poincaré rank does not decrease under gauge transformations.     

Number of lattice points in a sphere

A new estimate is obtained for the residue R in the asymptotics of the number of integer points in a ball of radius a. The estimate has the form R ? a ^{17/14 + ?} .     

Identifying variable points on a smooth curve

LetX be a compact Riemann surface,n ≥ 2 an integer andx = [x ₁, …,x _n ] an unorderedn-tuple of not necessarily distinct points onX. Byf _x :X →Y _x we denote the normalization which identifies thex ₁, …,x _n and maps them to the only and universal singularity of a complex curveY _x . Thenf _x depends holomorphically onx and is uniquely determined by this parameter. In this context we consider the fine moduli spaceQ _X of all complex-analytic quotients ofX and construct a morphismS ⁿ (X) →Q _X such that each and everyf _x corresponds to the image of the pointx on then-fold symmetric powerS ⁿ (X). For everyn ≥ 2 the mappingS ⁿ (X) →Q _X is a closed embedding; the points of its image have embedding dimensionn(n ? 1) inQ _X . HenceS ²(X) is a smooth connected component ofQ _X . On the other hand, a deformation argument yields thatS ⁿ (X) is part of the singular locus of the complex spaceQ _X provided thatn ≥ 3.