An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan

Displaying similar documents to “An iterative construction for ordinary and very special hyperelliptic curves”

An iterative construction for ordinary and very special hyperelliptic curves

Francis J. Sullivan (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si costruiscono famiglie di curve iperellittiche col $p$ —rango della varietà jacobiana uguale a zero. La costruzione sfrutta le proprietà elementari dell’operatore di Cartier e delle estensioni $p$ -cicliche dei corpi con la caratteristica $p$ maggiore di zero.

On ramified covers of the projective plane II: Generalizing Segre’s theory

Michael Friedman, Rebecca Lehman, Maxim Leyenson, Mina Teicher (2012)

Journal of the European Mathematical Society

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The classical Segre theory gives a necessary and sufficient condition for a plane curve to be a branch curve of a (generic) projection of a smooth surface in $ℙ^{3}$ . We generalize this result for smooth surfaces in a projective space of any dimension in the following way: given two plane curves, $B$ and $E$ , we give a necessary and sufficient condition for $B$ to be the branch curve of a surface $X$ in $ℙ^{N}$ and $E$ to be the image of the double curve of a $ℙ^{3}$ -model of $X$ . In the classical Segre theory, a...

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

Recovering an algebraic curve using its projections from different points. Applications to static and dynamic computational vision

Jeremy Yirmeyahu Kaminski, Michael Fryers, Mina Teicher (2005)

Journal of the European Mathematical Society

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We study some geometric configurations related to projections of an irreducible algebraic curve embedded in $ℂ ℙ^{3}$ onto embedded projective planes. These configurations are motivated by applications to static and dynamic computational vision. More precisely, we study how an irreducible closed algebraic curve $X$ embedded in $ℂ ℙ^{3}$ , of degree $d$ and genus $g$ , can be recovered using its projections from points onto embedded projective planes. The embeddings are unknown. The only input is the defining...

Nodal curves in $\mathbb{P}_{3}(\mathbb{C})$

Edoardo Ballico, Paolo Oliverio (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Siano $d$ , $g$ , $t$ interi con $0\leq t\leq g$ ; se esiste in $\mathbb{P}_{3}(\mathbb{C})$ una curva connessa, non singolare di grado $d$ e genere $g$ , allora esiste in $\mathbb{P}_{3}(\mathbb{C})$ una curva irriducibile di grado $d$ , genere aritmetico $g$ e $t$ nodi.

Frobenius nonclassicality with respect to linear systems of curves of arbitrary degree

Nazar Arakelian, Herivelto Borges (2015)

Acta Arithmetica

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For each integer s ≥ 1, we present a family of curves that are $_{q}$ -Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be $_{q}$ -Frobenius nonclassical with respect to the linear system of conics. In the $_{q}$ -Frobenius nonclassical cases, we determine the exact number of $_{q}$ -rational points. In the remaining cases, an upper bound for the number of $_{q}$ -rational points will follow from Stöhr-Voloch...

A curve of genus $q$ with a Half-Canonical embedding in $\mathbf{P}^{3}$

Sevin Recillas (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si costruiscono curve di genere $g=4n—3$ , $n\geq 3$ che hanno $2^{n-3}(2^{n-2}-1)$ fasci semicanonici $L$ tali che $h^{0}(L)=4$ . Per $n+3$ si dimostra che gli $L$ sono molto ampi.

The Mordell-Weil bases for the elliptic curve $y^{2} = x^{3} - m^{2} x + m^{2}$

Sudhansu Sekhar Rout, Abhishek Juyal (2021)

Czechoslovak Mathematical Journal

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Let $D_{m}$ be an elliptic curve over $ℚ$ of the form $y^{2} = x^{3} - m^{2} x + m^{2}$ , where $m$ is an integer. In this paper we prove that the two points $P_{- 1} = (- m, m)$ and $P_{0} = (0, m)$ on $D_{m}$ can be extended to a basis for $D_{m} (ℚ)$ under certain conditions described explicitly.

On invariants of elliptic curves on average

Amir Akbary, Adam Tyler Felix (2015)

Acta Arithmetica

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We prove several results regarding some invariants of elliptic curves on average over the family of all elliptic curves inside a box of sides A and B. As an example, let E be an elliptic curve defined over ℚ and p be a prime of good reduction for E. Let $e_{E} (p)$ be the exponent of the group of rational points of the reduction modulo p of E over the finite field $_{p}$ . Let be the family of elliptic curves $E_{a, b} : y^{2} = x^{3} + a x + b$ , where |a| ≤ A and |b| ≤ B. We prove that, for any c > 1 and k∈ ℕ, $1 / | | \sum_{E \in} \sum_{p \leq x} e_{E}^{k} (p) = C_{k} l i (x^{k + 1}) + O ((x^{k + 1}) / {(l o g x)}^{c}$ )as x → ∞, as long...

Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica (2020)

Czechoslovak Mathematical Journal

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We determine the distribution over square-free integers $n$ of the pair $({dim}_{𝔽_{2}} {Sel}^{Φ} (E_{n} / ℚ), {dim}_{𝔽_{2}} {Sel}^{\hat{Φ}} (E_{n}^{'} / ℚ))$ , where $E_{n}$ is a curve in the congruent number curve family, $E_{n}^{'} : y^{2} = x^{3} + 4 n^{2} x$ is the image of isogeny $Φ : E_{n} \to E_{n}^{'}$ , $Φ (x, y) = (y^{2} / x^{2}, y (n^{2} - x^{2}) / x^{2})$ , and $\hat{Φ}$ is the isogeny dual to $Φ$ .

Shells of monotone curves

Josef Mikeš, Karl Strambach (2015)

Czechoslovak Mathematical Journal

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We determine in $ℝ^{n}$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla$ for which any image of $C$ under a compact-free group $Ω$ of affinities containing the translation group is a geodesic with respect to $\nabla$ . Special attention is paid to the case that $Ω$ contains many dilatations or that $C$ is a curve in $ℝ^{3}$ . If $C$ is a curve in $ℝ^{3}$ and $Ω$ is the translation group then we calculate not only the components of the curvature and the Weyl...

On Zariski's theorem in positive characteristic

Ilya Tyomkin (2013)

Journal of the European Mathematical Society

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In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $- K_{S} . C + p_{g} (C) - 1$ , where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $𝚍𝚒𝚖 (V) = - K_{S} . C + p_{g} (C) - 1$ does not imply the nodality of $C$ even if $C$ belongs...