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    Algebraic curves violating the slope inequalities
    Osaka Journal of Mathematics (2015)
    Takao KatoGerriet Martens
    The gonality sequence $(d_{r})_{r \ge 1}$ of a curve of genus $g$ encodes, for $r 0$ such that there is no such jump, for all curves of genus $g$. Secondly, we compute the leading numbers (up to $r=19$) in the gonality sequence of an extremal space curve, i.e. of a space curve of maximal geometric genus w.r.t. its degree.
    Sequence (biology)
    Degree (music)
    Citations (1)
    An explicit algebraic family of genus-one curves violating the Hasse principle
    arXiv (Cornell University) (1999)
    Bjorn Poonen
    We prove that for any t in Q, the curve 5 x^3 + 9 y^3 + 10 z^3 + 12((t^12-t^4-1)/(t^12-t^8-1))^3 (x+y+z)^3 = 0 in P^2 is a genus 1 curve violating the Hasse principle. An explicit Weierstrass model for its Jacobian E_t is given. The Shafarevich-Tate group of each E_t contains a subgroup isomorphic to Z/3 x Z/3.
    Hasse principle
    Stable curve
    Hyperelliptic curve
    Citations (0)
    The Dimension of the Moduli Space of Pointed Algebraic Curves of Low Genus
    arXiv (Cornell University) (2022)
    Jan Stevens
    We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and determine the dimension for all semigroups of genus seven.
    Complex dimension
    Modular equation
    Citations (0)
    Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6
    arXiv (Cornell University) (2011)
    Michael Harrison
    We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.
    Degree (music)
    Citations (2)
    The maximum number of points on a curve of genus 4 over F_8 is 25
    arXiv (Cornell University) (2002)
    David SavittKristin Lauter
    This paper proves that the maximum number of rational points on a smooth, absolutely irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible using techniques from algebraic geometry. The appendix shows that 26 points is not possible by examining the zeta functions.
    Citations (1)
    The Geometric Bogomolov Conjecture for Small Genus Curves
    arXiv (Cornell University) (2008)
    Xander Faber
    The Bogomolov Conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov Conjecture for all curves of genus at most 4 defined over a function field of characteristic zero. We recover the known result for genus 2 curves and in many cases improve upon the known bound for genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved.
    Citations (10)
    Explicit Solution By Radicals, Gonal Maps and Plane Models of Algebraic Curves of Genus 5 or 6
    arXiv (Cornell University) (2011)
    Michael Harrison
    We give explicit computational algorithms to construct minimal degree (always $\le 4$) ramified covers of $\Prj^1$ for algebraic curves of genus 5 and 6. This completes the work of Schicho and Sevilla (who dealt with the $g \le 4$ case) on constructing radical parametrisations of arbitrary genus $g$ curves. Zariski showed that this is impossible for the general curve of genus $\ge 7$. We also construct minimal degree birational plane models and show how the existence of degree 6 plane models for genus 6 curves is related to the gonality and geometric type of a certain auxiliary surface.
    Degree (music)
    Citations (0)
    The Maximum Number of Points on a Curve of Genus 4 over is 25
    Canadian Journal of Mathematics (2003)
    David Savitt
    Abstract We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.
    Citations (6)
    ON THE GENUS OF AN ALGEBRAIC CURVE A(x) − B(y) = 0
    arXiv (Cornell University) (2015)
    Fedor Pakovich
    We show that for a fixed rational function A ∈ C(z) the genus of an algebraic curve A(x) − B(y) = 0, where B ∈ C(z), tends to infinity as deg B tends to infinity, unless there exists a compact Riemann surface R of genus 0 or 1 and a regular covering f : R → CP1 such that f = A◦p for some p : R → CP 1 .
    Infinity
    Compact Riemann surface
    Citations (3)
    The Geometric Bogomolov Conjecture for Curves of Small Genus
    Experimental Mathematics (2009)
    Xander Faber
    Abstract The Bogomolov conjecture is a finiteness statement about algebraic points of small height on a smooth complete curve defined over a global field. We verify an effective form of the Bogomolov conjecture for all curves of genus at most 4 over a function field of characteristic zero. We recover the known result for genus-2 curves and in many cases improve upon the known bound for genus-3 curves. For many curves of genus 4 with bad reduction, the conjecture was previously unproved. Keywords: Primary 11G30Secondary 14G40, 11G50Bogomolov conjecturecurves of higher genusfunction fieldsmetric graphs
    Function field
    Zero (linguistics)
    Citations (22)