Displaying similar documents to “Lang's Conjectures, Fibered Powers, and Uniformity.”

Nagata submaximal curves on ℙ¹ × ℙ¹

Wioletta Syzdek (2003)

Annales Polonici Mathematici

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The aim of this paper is to show that on ℙ¹ × ℙ¹ with a polarization of type (2,1) there are no R-R expected submaximal curves through any 10 ≤ r ≤ 15 points.

A note on the plane Jacobian conjecture

Nguyen Van Chau (2012)

Annales Polonici Mathematici

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It is shown that every polynomial function P:ℂ² → ℂ with irreducible fibres of the same genus must be a coordinate. Consequently, there do not exist counterexamples F = (P,Q) to the Jacobian conjecture such that all fibres of P are irreducible curves with the same genus.

Generalized Mukai conjecture for special Fano varieties

Marco Andreatta, Elena Chierici, Gianluca Occhetta (2004)

Open Mathematics

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Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension...

Lang-Trotter and Sato-Tate distributions in single and double parametric families of elliptic curves

Min Sha, Igor E. Shparlinski (2015)

Acta Arithmetica

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We obtain new results concerning the Lang-Trotter conjectures on Frobenius traces and Frobenius fields over single and double parametric families of elliptic curves. We also obtain similar results with respect to the Sato-Tate conjecture. In particular, we improve a result of A. C. Cojocaru and the second author (2008) towards the Lang-Trotter conjecture on average for polynomially parameterised families of elliptic curves when the parameter runs through a set of rational numbers of...