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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 14H: Algebraic Curves

Introduction

History

Applications and related fields

Subfields

• 14H05: Algebraic functions; function fields, See also 11R58
• 14H10: Families, moduli (algebraic)
• 14H15: Families, moduli (analytic), See also 30F10, 32Gxx
• 14H20: Singularities, local rings, See also 13Hxx
• 14H25: Arithmetic ground fields, See also 11Dxx, 11G05, 14Gxx
• 14H30: Coverings, fundamental group, See also 14E20, 14F35
• 14H37: Automorphisms [new in 2000]
• 14H40: Jacobians, See also 32G20
• 14H42: Theta functions; Schottky problem, See also 14K25, 32G20
• 14H45: Special curves and curves of low genus
• 14H50: Space curves
• 14H51: Special divisors (gonality, Brill-Noether theory) [new in 2000]
• 14H52: Elliptic curves. See also 11G05, 11G07, 14Kxx
• 14H55: Riemann surfaces; Weierstrass points; gap sequences, See also 30Fxx
• 14H60: Vector bundles on curves, See also 14F05
• 14H70: Relationships with integrable systems [new in 2000]
• 14H81: Relationships with physics [new in 2000]
• 14H99: None of the above but in this section

Parent field: 14 - Algebraic geometry

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Software and tables

• A compendium of plane curves
• Notice of software for computing zeta functions of (projective) curves.

Other web sites with this focus

Selected topics at this site

• What is Bezout's theorem (and who was Bezout?)
• How do you parameterize a curve -- i.e. how do you know that a quadratic curve in 2 variables with a rational point is in one-to-one correspondence with the rationals? (It is a rational variety)
• How many points on a conic over F_p?
• Characterization of equivalence classes of quadratic curves.
• Solving quadratic equations over Q and Z. (that is, studying rational conic curves).
• Finding all integral solutions to a homogeneous quadratic in 3 variables -- example.
• Solutions to x^3 + y^2 = z^2 ?
• What if a curve is not parameterizable -- just how simple can you make it?
• Finding a curve of minimal degree in the plane which passes through a given set of points (an answer does appear, toward the end!)
• [Offsite] Constructing curves with many rational points (curves of a fixed genus g bigger than 1) (N.B. - there can't be too many) [Noam Elkies]
• Calculating the envelope around a curve (a new curve a fixed perpendicular distance away).
• What's the distance between a point and a parabola? (This is essentially an elimination-theoretic description of an envelope of the parabola.)
• Recognizing a curve of genus 0
• Rational solutions to x^4 + y^4 = 17
• Solutions to y^2 = x^m + 1 (Catalan)
• Current status: consecutive proper powers -- probably only {8,9} (Catalan)
• Solutions to y^q=1+x+x^2+...+x^r (Ljunggren)
• Some curves of high genus: rational solutions to y^k = f_k(x) where f_k(tan(u)) = tan(k u)/tan(u)
• What is known about the density of rational points in algebraic curves (as opposed to "mere" infinitude)?
• Sets of points on the unit circle with rational interpoint distances