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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 14J: Surfaces and higher-dimensional varieties

Introduction

History

Applications and related fields

Subfields

• 14J10: Families, moduli, classification: algebraic theory
• 14J15: Moduli, classification: analytic theory, See also 32G13, 32J15
• 14J17: Singularities of surfaces
• 14J20: Arithmetic ground fields, See also 11Dxx, 11G25, 11G35, 14Gxx
• 14J25: Special surfaces, For Hilbert modular surfaces, See 14G35
• 14J26: Rational and ruled surfaces
• 14J27: Elliptic surfaces
• 14J28: K3 surfaces and Enriques surfaces
• 14J29: Surfaces of general type
• 14J30: [Special] 3-folds, See also 14E05
• 14J32: Calabi-Yau manifolds, mirror symmetry [new in 2000]
• 14J35: [Special] 4-folds, See also 14E05
• 14J40: [Special] n-folds (n > 4)
• 14J45: Fano varieties
• 14J50: Automorphisms of surfaces and higher-dimensional varieties. See also 14E09
• 14J60: Vector bundles on surfaces and higher-dimensional varieties. See also 14F05, 32Lxx
• 14J70: Hypersurfaces
• 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants) [new in 2000]
• 14J81: Relationships with physics [new in 2000]
• 14J99: 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

Other web sites with this focus

Selected topics at this site

• Describing the family of reducible cubic surfaces among all cubic surfaces (hopeless?)
• What can an algebraic surface in R^3 look like?
• What can a quadratic surface in R^3 look like?
• Name this cubic surface!
• A surface may be presented as fibred over a curve, with fibres of different genera.
• How can you tell if an algebraic surface is rational over a specific field?
• There are infinitely many rational points on the Euler ("Fermat") surface x^4+y^4+z^4=1; is there a parameterized family? (Open)
• Solutions to x^3 + y^3 = z^2
• Sum of two fifth powers a square?
• Small generalization of FLT: x^n + y^n = z^m has solutions iff gcd(n,m)=1 or n=m=2.
• The Beal Prize conjecture: solutions to x^n + y^m = z^r
• Solutions to generalized Fermat equation x^a+y^b=z^c.
• Examples of pairs of perfect powers which sum to another perfect power.
• There are infinitely many solutions to x^2 + y^3 = z^4
• Solutions to A x^p + B y^q = C z^r? (none with A=B=C=1 if p,q,r greater than 2)
• Which is larger: a^(1/3) or b^(1/3)+c^(1/3) (remarkable close calls are solutions to |(a-b-c)^3-27abc|=1)
• Parameterizing the solutions to x^3+y^3+z^3+w^3=0
• Solutions to x^3 + y^3 = z^3 + 1
• Solving the pair x^2 + y^2 = u^4, x+y = v^2 in integers.
• Seeking integral points on the intersections of 3 quadratics in P^4.
• Triangles with integer sides and area, two sides being consecutive squares
• The spider-on-a-box problem: there are parameterized families of integer boxes with all three geodesic distances between opposite corners being integral too.
• Three 120 degree angles inside a triangle, all six lengths integral
• Yet another elliptic surface
• Three rational right triangles on the same hypotenuse whose areas are in arithmetic progression?
• Three rational right triangles on the same hypotenuse whose areas sum
• Arithmetic progression of four terms nearly square
• A few near misses of solutions to the Integral Brick problem.
• The rational box: still open
• Update on searches for a rational box.
• Rational boxes with weakened conditions to be met
• Computer searches for the Integer Cuboid - an update
• Construct a three-by-three magic square with square entries? (open)
• No nontrivial solutions known for x^5 + y^5 + z^5 + t^5 = 0
• How many normals to a surface meet at a point?
• Parameterizing the family of lines tangent to two spheres (an algebraic surface).
• "Freshman addition of reciprocals": 1/(A+B+C+D) = (1/A) + (1/B) + (1/C) + (1/D)
• The hyperbolic Pythagorean theorem, a^2 + b^2 = c^2 * (1+a^2*b^2)
• Two squares from two numbers: finding a and b so that a+b^2 and b+a^2 are both squares.
• Four squares from three integers: can ab+1, bc+1, ca+1, and abc+1 all be squares?