What is Algebraic Geometry?

The Barth sextic surface
with 65 nodes
Courtesy W. Barth (Erlangen) & S. Endrass (Mainz)

Algebraic Geometry Group

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Algebraic geometry is the study of the "shape" of the set of solutions to polynomial equations. For example, the set of solutions to the single equation:

5x - 3y = 12


is a straight line in the ( x, y)-plane, but a single polynomial equation of higher degree, like

x2 + 3x - y2 = 12


has a solution made up of the two arms of a hyperbola, or, if we add two "ideal points" at infinity, the solution set becomes a closed loop.

Things get more complicated when the degree of the equation is three or more. One can get many closed loops, some inside others, etc., and even at this "simple" level the complete classification of what configurations can occur for polynomials of degree n is still a mystery.

In another direction, one can ask when equations like the above have solutions with both  x and  y whole numbers (integers) or with both  x and  y fractions (rational numbers). This again becomes very hard when the degree of the polynomial is three or more. In fact, the recent celebrated proof by Andrew Wiles of Fermat's last theorem is just one very special case, namely the equation

xn + yn = 1


has no fractional solutions when  n is 3 or more. The study of this type of question is called "arithmetic algebraic geometry" and is closely related to number theory, group theory, and representation theory.

When there are more than two variables and/or two or more polynomials, again the situation is far from completely understood, even without worrying about whether the solutions are rational numbers. This general study is called "higher dimensional algebraic geometry" and the best understood case is when the solutions are complex numbers. In this case the solutions form a (generally) smooth continuum whose dimension is expected to be the number of variables minus the number of polynomials. The study of the geometric properties of this continuum is known as "complex algebraic geometry" and is closely related to topology, differential geometry, complex analysis and even theoretical physics.

Several researchers in the math department at Utah work in the areas of arithmetic algebraic geometry and complex algebraic geometry.