IV.4 Algebraic Geometry János Kollár

i IV.4. Algebraic Geometry 363 IV.4 Algebraic Geometry János Kollár 1 Introduction Succinctly put, algebraic geometry is the study of geometry using polynomials and the investigation of polynomials using geometry. Many of us were taught the beginnings of algebraic geometry in high school, under the name “analytic geometry.” When we say that y mx b is the equa- = + tion of a line L, or that x2 y2 r 2 describes a circle C + = of radius r , we establish a basic connection between geometry and algebra. If we want to find the points where the line L and Figure 1 A hyperboloid intersecting a plane. the circle C intersect, we just substitute mx b for + y in the circle equation to get x2 (mx b)2 r 2 + + = sets: L is the zero set of y mx b (that is, the set of and solve the resulting quadratic equation to obtain the − − all points (x, y) such that y mx b 0) and C is the x coordinates of the two intersection points. − − = zero set of x2 y2 r 2. This simple example encapsulates the method of + − Similarly, the zero set of 2x2 3y2 z2 7 in 3-space algebraic geometry: a geometric problem is translated + − − is a hyperboloid, the zero set of z x y in 3-space is a into algebra, where it is readily solvable; conversely, we − − plane, and the common zero set of these two equations get insight into algebra problems by using geometry. in 3-space is the intersection of the hyperboloid and the It is hard to guess the solutions of systems of poly- plane, which is an ellipse (see figure 1). nomial equations, but once a corresponding geometric The set of common zeros of a system of polyno- picture is drawn, we start to have a qualitative under- mial equations in any number of variables is called an standing of them. The precise quantitative answer is algebraic set. These are the basic objects of algebraic then provided by algebra. geometry. Most people feel that geometry ends in 3-space. Very 2 Polynomials and Their Geometry few have a feeling for 4-space, also called space-time, Polynomials are the expressions one can put together and 5-space is by and large inconceivable to almost from variables and numbers by addition and multipli- everyone. So what is the meaning of geometry in many cation. The most familiar are one-variable polynomials variables? such as x3 x 4, but we can use two or three vari- Algebra comes to our rescue here. While I have great − + ables to get, for instance, 2x5 3xy2 y3 (which has difficulty visualizing what a four-dimensional sphere of − + degree 5 in two variables) or x5 y7 x2z8 xyz 1 radius r in 5-space should be, I can easily write down − + − + (which has degree 10 in three variables). In general, one its equation, can use n variables, in which case they are frequently 2 2 2 2 2 2 x1 x2 x3 x4 x5 r 0, denoted by x1,x2,...,xn, and we write f (x1,...,xn), + + + + − = f(x) or simply f to denote an unspecified polynomial. and work with it. This equation is also something a Polynomials are the only functions that computers computer can handle, which is immensely useful in can work with. (Although your pocket calculator is applications. likely to have a button for logarithms, it is secretly com- I will, nonetheless, stick to two or three variables puting a polynomial whose value at a number b agrees for the rest of this article. This is where all geometry with log b up to many decimal places.) starts and there are plenty of interesting questions and We can slightly rewrite the equations we gave earlier results. for the line L and the circle C: as y mx b 0 and The importance of algebraic geometry derives from − − = x2 y2 r 2 0. We can then describe L and C as zero the fact that significant interactions between algebra + − = i 364 IV. Branches of Mathematics and geometry happen very frequently. Let us look at Let us now look at the same equation, but declare that two examples, just for illustration. we care only about the parities of the two sides (that is, whether they are even or odd). For instance, 32 152 and + 3 Most Shapes Are Algebraic 42 are both even, so we say that 32 152 42 (mod 2). + ≡ The parities of x2 y2 and of z2 depend only on those Shapes that occur frequently enough to have their own + of x, y, and z, so we can pretend that x, y, and z are all name, for instance, lines, planes, circles, ellipses, hyper- either 0 (the even case) or 1 (the odd case). Our equation bolas, parabolas, hyperboloids, paraboloids, ellipsoids, modulo 2 therefore has four solutions: are almost all algebraic. Even the more esoteric con- choid (or shell curve) of Dürer, the trident of newton 000, 011, 101, 110. [VI.14], and the folium of Kepler are algebraic. These look like code words in a computer message. Some shapes cannot be described by polynomial It was quite a surprise when it was discovered that equations, but they can be described by polynomial using polynomials and their solutions modulo 2 is a inequalities. For instance, the inequalities 0 ! x ! a great—probably the best—way of constructing error- and 0 together describe a rectangle with side ! y ! b correcting codes [VII.6 §§3–5]. lengths a, b. Shapes described by polynomial inequal- There is something very substantial and new happen- ities are called semialgebraic, and every polyhedron is ing here. Let us think for a moment about what 3-space semialgebraic. is for us. For many it is an amorphous everything, but Not everything is an algebraic set, though. Look, for for algebraic geometers (with descartes [VI.11] as our example, at the graph of the sine function y sin x. = ancestor) it is simply a collection of points described This crosses the x-axis infinitely many times (at multi- by three numbers, the x, y, and z coordinates. Let us ples of π). If f (x) is any polynomial, then it has at most make a jump here, and declare that “3-space modulo 2” as many roots as its degree, so y f (x) will never look = is the collection of all “points” given by three coordin- like y sin x. = ates modulo 2. Four of these are listed above, and there We can, however, get very close to sin x with a poly- are four more. The beauty of algebra is that suddenly nomial if we concentrate on values of x that are not too we can talk about lines, planes, spheres, cones in this large. For instance, the degree-7 Taylor polynomial “3-space having only eight points.” 1 1 1 x x3 x5 x7 − 6 + 120 − 5040 We do not need to stop here, and one can work mod- differs from sin x by an error of at most 0.1 for π< ulo any integer. For example, working modulo 7, we − x<π. This is a very special case of a basic theo- have 0, 1, 2, 3, 4, 5, 6 as possible coordinates, and so rem of Nash that says that every “reasonable” geomet- “3-space modulo 7” has 73 343 points. = ric shape is algebraic if we ignore what happens very Talking about geometry in these spaces is very far from the origin. So, what is reasonable? Certainly intriguing, but also technically difficult. Its great reward not everything. Fractals seem profoundly nonalgebraic. is that one can view this process as a “discretization” The nicest shapes are manifolds [I.3 §6.9], and all of of ordinary space. Working modulo n for large n (espe- these can be described by polynomials. cially when n is a prime number) gets very close to the usual geometry. Nash’s theorem. Let M be any manifold in Rn. Fix any This approach is especially fruitful in number-theo- large number R. Then there is a polynomial f whose retic questions. It was, for instance, instrumental in zero set is as close to M as we want, at least inside a Wiles’s proof of Fermat’s last theorem. ball of radius R around the origin. For more on these topics, see arithmetic geometry 4 Codes and Finite Geometries [IV.5]. Consider the equation x2 y2 z2, which describes + = 5 Snapshots of Polynomials a double cone in 3-space (see figure 4). If we confine ourselves to natural numbers, then the solutions of Consider the equation x2 y2 R. If R>0, then the + = x2 y2 z2 are the Pythagorean triples, correspond- real solutions form a circle of radius √R; if R 0, we + = = ing to right-angled triangles where all sides have inte- get only the origin; and if R<0, we get the empty set. ger lengths, of which the two best-known examples are Thus, if R>0, then the geometry of the solution set (3, 4, 5) and (5, 12, 13). determines what R is, but otherwise it does not. We i IV.4. Algebraic Geometry 365 can of course look at complex solutions, and the com- Hilbert’s Nullstellensatz. Two complex polynomials f plex solutions always determine R. (For instance, the and g have the same complex solutions if and only if intersection points with the x-axis are ( √R, 0).) they have the same irreducible factors. ± If R is a rational number, we can ask about rational We can do even better for polynomials with integer 2 2 solutions of x y R, and if R is an integer, we coefficients.

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