Displaying similar documents to “An explicit algebraic family of genus-one curves violating the Hasse principle”

On the gonality of curves in 𝐏 n

Edoardo Ballico (1997)

Commentationes Mathematicae Universitatis Carolinae

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Here we study the gonality of several projective curves which arise in a natural way (e.gċurves with maximal genus in 𝐏 n , curves with given degree d and genus g for all possible d , g if n = 3 and with large g for arbitrary ( d , g , n ) ).

Some examples of Gorenstein liaison in codimension three.

Robin Hartshorne (2002)

Collectanea Mathematica

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Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension 2 in projective space. In this paper we study points in P3 and curves in P4 in an attempt to see how far typical codimension 2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. This examples are candidates for counterexamples to the hoped-for extensions...

A group law on smooth real quartics having at least 3 real branches

Johan Huisman (2002)

Journal de théorie des nombres de Bordeaux

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Let C be a smooth real quartic curve in ℙ 2 . Suppose that C has at least 3 real branches B 1 , B 2 , B 3 . Let B = B 1 × B 2 × B 3 and let O ∈ B . Let τ O be the map from B into the neutral component Jac ( C ) ( ℝ ) 0 of the set of real points of the jacobian of C , defined by letting τ O ( P ) be the divisor class of the divisor ∑ P i - O i . Then, τ O is a bijection. We show that this allows an explicit geometric description of the group law on Jac ( C ) ( ℝ ) 0 . It generalizes the classical geometric description of the group law on the neutral component of the set of real...

Curves of genus seven that do not satisfy the Gieseker-Petri theorem

Abel Castorena (2005)

Bollettino dell'Unione Matematica Italiana

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In the moduli space of curves of genus g , ℳ g , let 𝒢𝒫 g be the locus of curves that do not satisfy the Gieseker-Petri theorem. In the genus seven case we show that 𝒢𝒫 7 is a divisor in ℳ 7 .

Vector bundles on plane cubic curves and the classical Yang–Baxter equation

Igor Burban, Thilo Henrich (2015)

Journal of the European Mathematical Society

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In this article, we develop a geometric method to construct solutions of the classical Yang–Baxter equation, attaching a family of classical r -matrices to the Weierstrass family of plane cubic curves and a pair of coprime positive integers. It turns out that all elliptic r -matrices arise in this way from smooth cubic curves. For the cuspidal cubic curve, we prove that the obtained solutions are rational and compute them explicitly. We also describe them in terms of Stolin’s classication...