Invariants of real symplectic four-manifolds out of reducible and cuspidal curves

Jean-Yves Welschinger

Displaying similar documents to “Invariants of real symplectic four-manifolds out of reducible and cuspidal curves”

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...

Welschinger invariants of small non-toric Del Pezzo surfaces

Ilia Itenberg, Viatcheslav Kharlamov, Eugenii Shustin (2013)

Journal of the European Mathematical Society

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We give a recursive formula for purely real Welschinger invariants of the following real Del Pezzo surfaces: the projective plane blown up at $\sim q$ real and $s \leq 1$ pairs of conjugate imaginary points, where $q + 2 s \leq 5$ , and the real quadric blown up at $s \leq 1$ pairs of conjugate imaginary points and having non-empty real part. The formula is similar to Vakil’s recursive formula [22] for Gromov–Witten invariants of these surfaces and generalizes our recursive formula [12] for purely real Welschinger invariants...

Rational Points on Certain Hyperelliptic Curves over Finite Fields

Maciej Ulas (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation $S_{k}^{i} : u ² = \prod_{j = 1}^{k} g_{i} (x_{j})$ , i=1,2, contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves $C_{i} : y ² = g_{i} (x)$ , (i=1,2) defined over a finite field, in polynomial time.

The automorphism group of ${\bar{M}}_{0, n}$

Andrea Bruno, Massimiliano Mella (2013)

Journal of the European Mathematical Society

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The paper studies fiber type morphisms between moduli spaces of pointed rational curves. Via Kapranov’s description we are able to prove that the only such morphisms are forgetful maps. This allows us to show that the automorphism group of ${\bar{M}}_{0, n}$ is the permutation group on $n$ elements as soon as $n \geq 5$ .

The Analytic Rank of a Family of Jacobians of Fermat Curves

Tomasz Jędrzejak (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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We study the family of curves $F_{m} (p) : x^{p} + y^{p} = m$ , where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves $F_{m} (p)$ . As a corollary we conclude that the jacobians of the curves $F_{m} (5)$ with even analytic rank and those with odd analytic rank are equally distributed.

Labeled floor diagrams for plane curves

Sergey Fomin, Grigory Mikhalkin (2010)

Journal of the European Mathematical Society

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Floor diagrams are a class of weighted oriented graphs introduced by E. Brugallé and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative) Gromov–Witten invariants of projective spaces in terms of floor diagrams and their generalizations. In a number of cases, these descriptions can be used to obtain explicit (direct or recursive) formulas for the corresponding enumerative invariants. In particular, we use this approach to enumerate...

Optimal curves differing by a 5-isogeny

Dongho Byeon, Taekyung Kim (2014)

Acta Arithmetica

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For i = 0,1, let $E_{i}$ be the $X_{i} (N)$ -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational...

Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve $C$ over $ℚ$ . The focus is on practical aspects of this problem in the case that the genus of $C$ is at least $2$ , and therefore the set of rational points is finite.

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 Lincei. Matematica e Applicazioni

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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.

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Janusz Gwoździewicz, Arkadiusz Płoski (2005)

Colloquium Mathematicae

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For every polynomial F in two complex variables we define the Łojasiewicz exponents $ℒ_{p, t} (F)$ measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents $ℒ_{p, t} (F)$ in terms of the local invariants of singularities of the pencil of projective curves associated with F.

An index inequality for embedded pseudoholomorphic curves in symplectizations

Michael Hutchings (2002)

Journal of the European Mathematical Society

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Let $Σ$ be a surface with a symplectic form, let $φ$ be a symplectomorphism of $Σ$ , and let $Y$ be the mapping torus of $φ$ . We show that the dimensions of moduli spaces of embedded pseudoholomorphic curves in $ℝ \times 𝕐$ , with cylindrical ends asymptotic to periodic orbits of $φ$ or multiple covers thereof, are bounded from above by an additive relative index. We deduce some compactness results for these moduli spaces. This paper establishes some of the foundations for a program with Michael Thaddeus, to...

A remark on a conjecture of Hain and Looijenga

Carel Faber (2011)

Annales de l’institut Fourier

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We show that the natural generalization of a conjecture of Hain and Looijenga to the case of pointed curves holds for all $g$ and $n$ if and only if the tautological rings of the moduli spaces of curves with rational tails and of stable curves are Gorenstein.

Complete pluripolar curves and graphs

Tomas Edlund (2004)

Annales Polonici Mathematici

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It is shown that there exist $C^{\infty}$ functions on the boundary of the unit disk whose graphs are complete pluripolar. Moreover, for any natural number k, such functions are dense in the space of $C^{k}$ functions on the boundary of the unit disk. We show that this result implies that the complete pluripolar closed $C^{\infty}$ curves are dense in the space of closed $C^{k}$ curves in ℂⁿ. We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar. ...

Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents

Jun-Muk Hwang (2010)

Annales scientifiques de l'École Normale Supérieure

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We formulate the equivalence problem, in the sense of É. Cartan, for families of minimal rational curves on uniruled projective manifolds. An important invariant of this equivalence problem is the variety of minimal rational tangents. We study the case when varieties of minimal rational tangents at general points form an isotrivial family. The main question in this case is for which projective variety $Z$ , a family of minimal rational curves with $Z$ -isotrivial varieties of minimal rational...

Some examples of 5 and 7 descent for elliptic curves over $Q$

Tom Fisher (2001)

Journal of the European Mathematical Society

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We perform descent calculations for the families of elliptic curves over $Q$ with a rational point of order $n = 5$ or 7. These calculations give an estimate for the Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over $Q$ may become arbitrarily large.

Tautological relations and the $r$ -spin Witten conjecture

Carel Faber, Sergey Shadrin, Dimitri Zvonkine (2010)

Annales scientifiques de l'École Normale Supérieure

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In [11], A. Givental introduced a group action on the space of Gromov–Witten potentials and proved its transitivity on the semi-simple potentials. In [24, 25], Y.-P. Lee showed, modulo certain results announced by C. Teleman, that this action respects the tautological relations in the cohomology ring of the moduli space ${\bar{ℳ}}_{g, n}$ of stable pointed curves. Here we give a simpler proof of this result. In particular, it implies that in any semi-simple Gromov–Witten theory where arbitrary correlators...