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[Texts]## 14H: Algebraic Curves |

- 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.

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

- 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

Last modified 2000/01/14 by Dave Rusin. Mail: