Hypersurfaces with too many rational curves

We study smooth hypersurfaces of degree \(d\ge n+1\) in \(\mathbf{P}^n\) whose spaces of smooth rational curves of low degrees are larger than expected, and show that under certain conditions, the primitive part of the middle cohomology of such hypersurfaces have non-trivial Hodge substructures. As an application, we prove that the space of lines on any smooth Fano hypersurface of degree \(d \le 8\) in \(\mathbf{P}^n\) has the expected dimension \(2n-d-3\) .     

Shimura curves with many uniform dessins

A compact Riemann surface of genus g >?1 has different uniform dessins d’enfants of the same type if and only if its surface group S is contained in different conjugate Fuchsian triangle groups Δ and αΔα ^?1. Tools and results in the study of these conjugates depend on whether Δ is an arithmetic triangle group or not. In the case when Δ is not arithmetic the possible conjugators are rare and easy to classify. In the arithmetic case, i.e. for Shimura curves, the problem is much more complicated, but the arithmetic of quaternion algebras controls the growth of the number of uniform dessins of given type with respect to the genus. This number grows at most as O(g ^1/3) and this bound is sharp. As a tool, localization of the quaternion algebras and the representation of p-adic maximal orders as vertices of Serre–Bruhat–Tits trees turn out to be crucial. In low genera, the results shed a surprising new light on the uniformization of some classical curves like Klein’s quartic and other Macbeath–Hurwitz curves.     

Trigonometric polynomials with many real zeros

This is a contribution to the theory of “incomplete trigonometric polynomials”T _n , but mainly for the case when their zeros are not concentrated at just one point, but are distributed in some intervalI whose length is not too large. We begin with the simple theorem that if ∥T _n ∥ ≤ 1 and ifT _n has ≥θn, 0<θ< 2, zeros at 0, thenT _n (t) must be small on the interval |t|<2 arcsin (θ/2). There are similar but more complicated and more difficult to prove results whenT _n has ≥θn zeros onI. These results have the following application: IfT _n →f a.e., and if ∥T _n >∥_∞<-1, thenf vanishes on a set of the circleT whose measure is controlled by lim sup (N _n /n), whereN _n is the number of zeros ofT _n onT. In turn, this has further applications to series of polynomials, to norms of Lagrange operators, and to Hardy classes.     

Enriched spin curves on stable curves with two components

In [7], Mainò constructed a moduli space for enriched stable curves, by blowing-up the moduli space of Deligne–Mumford stable curves. We introduce enriched spin curves, showing that a parameter space for these objects is obtained by blowing-up the moduli space of spin curves. The author was partially supported by CNPq (Proc.151610/2005-3) and by Faperj (Proc.E-26/152-629/2005).     

Topology of real algebraic curves

The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.     

On minimizing partitions with infinitely many components

It is known [8] that, wheng ∈L ⁿ (Ω) (Ω open and bounded inR ⁿ , with ≪regular≫ boundary∂Ω), any minimizer (K, w) of the functional among relatively closed subsetsC ofΩ and piecewise-constant functionsu onΩ/C, gives rise to a finite decomposition ofΩ/K. Here we exhibit a piecewise-constant functiong on the unit diskD ofR ², with radial symmetry, such thatg ∈L ^q (D) for all 1 ⩽q < 2 and the unique minimizer of F has infinitely many components. We also fill a gap occurred in the proof of Proposition 5.2 of [8].

---

Sunto è noto [8] che quandog ∈L ⁿ (Ω (Ω aperto limitato diR ⁿ , con frontiera sufficientemente regolare) i minimi (K, w) del funzionale , doveC è relativamente chiuso in Ω eu è costante a tratti suΩ/C, danno luogo a decomposizioni finite diΩ/K. In questo lavoro mostriamo un controesempio relativo ad un datog ∈L ^q (D) per ogni 1 ⩽q < 2 (D è il disco unitario diR ²), a simmetria radiale e costante a tratti. Viene inoltre corretto un errore occorso nella dimostrazione della Prop. 5.2 di [8].

     

On the number of connected components of real abelian varieties that admit sufficiently many complex multiplications

Supported by the Netherlands Organisation for Scientific Research (NWO)     

The genus of curves over finite fields with many rational points

We prove the following result which was conjectured by Stichtenoth and Xing: letg be the genus of a projective, irreducible non-singular algebraic curve over the finite field and whose number of -rational points attains the Hasse-Weil bound; then either 4g≤(q−1)² or 2g=(q−1)q. Supported by a grant from the International Atomic Energy and UNESCOCorrespondence to: F. Torres This article was processed by the author using theLatex style file from Springer-Verlag.     

p-Torsion in the class group of curves with too many automorphisms

Archiv der Mathematik -     

On the topological classification of rarefaction curves in systems of three conservation laws

In this paper we study the topological properties of integral curves of a system of implicit differential equations associated to rarefaction curves of a system of three conservation laws. This system of equations becomes singular at the points of eigenvalue of multiplicity greater or equal to two. We focus our attention to the generic case of multiplicity two and three. We give local weak topological models for these equations.     

Limit canonical systems on curves with two components

In the 80’s D. Eisenbud and J. Harris considered the following problem: “What are the limits of Weierstrass points in families of curves degenerating to stable curves?” But for the case of stable curves of compact type, treated by them, this problem remained wide open since then. In the present article, we propose a concrete approach to this problem, and give a quite explicit solution for stable curves with just two irreducible components meeting at points in general position. Oblatum 15-VIII-2000 & 8-I-2001?Published online: 9 April 2002     

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k(R) be the smallest degree of a pencil on X (the real gonality of X). Then we can find on X a base point free pencil of degree g+1 (resp. g if X is not hyperelliptic, i.e. if k(R)>2) with an assigned geometric behaviour w.r.t. the real components of X, and if we prove that which is the same bound as for the gonality of a complex curve of even genus g. Furthermore, if the complexification of X is a k-gonal curve (k≥2) one knows that k≤k(R)≤2k−2, and we show that for any two integers k≥2 and 0≤n≤k−2 there is a real curve with real points and k-gonal complexification such that its real gonality is k+n.     

Clifford's inequality for real algebraic curves

We improve Clifford's Inequality for real algebraic curves. As an application we improve Harnack's Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.