I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.
So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.
The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set of polynomials in some polynomial ring over some field . And we write for the set of all simultaneous roots to all these polynomials:
If we write our polynomials as coming from the ring , then the corresponding solution points will be points in the vector space . In order to emphasize that we do not care for the vector space structure of this space, we shall denote it with , or if we want to emphasize the field, with .
The first observation at this point is that if we take the polynomial , then the solution set over is empty, while the solution set over is not. So, in order to set all solution sets on an equal footing – and also to make the later occurring correspondences work out – we shall require to be an algebraically closed field. In other words, we can always find a root to any polynomial.
We call the solution sets varieties (or – in order to distinguish from everything else we might encounter, we shall call them affine algebraic varieties).
So, the study of solutions to systems of polynomial equations is the study of varieties. And hence geometry. This neatly expands on the classical linear algebra viewpoint – where we study systems of linear equations as intersections of planes. It turns out that the main computational approach – GrŲbner bases – actually specializes to the GauŖ algorithm on matrices if we specialize to linear systems of equations.
The unit circle is an affine variety, given by the vanishing of . The hyperbola and the parabola are also affine varieties.
Another very commonly used example is the twisted cubic curve – given by the simultaneous vanishing of the parabolic sheet and the cubic curve sheet .
Changing the defining equations
The points p at which both f and g vanish are the same points as those where f and f+g vanish. Indeed, if f(p)=0 and g(p)=0, then (f+g)(p)=0 as well, so the points in V(f,g) are all contained in V(f,f+g). On the other hand, suppose that f(p)=0 and (f+g)(p)=0. Then 0=(f+g)(p)=f(p)+g(p)=0+g(p). So g(p)=0 as well. Hence, all points in V(f,f+g) are contained in V(f,g). So the two varieties are equal.
Suppose that . Then for any h=fg we’ll see that h(p)=f(p)g(p)=0g(p)=0. Hence, if f vanishes at p, then every polynomial that f divides will also vanish at p.
Now, a set of polynomials in that is closed under addition and multiplication by elements from is an ideal of the polynomial ring. Thus, these considerations convince us that the set of all polynomials in that all vanish on all points in is in fact an ideal.
So we can define the variety from an ideal – given , we write V(I) for the set of all points such that f(p)=0 for all .
The sum of two ideals I and J are the set of all elements on the form f+g where f is from I and g is from J. If I is generated by and J is generated by , then I+J has generators . And the set of points where all the polynomials in I+J vanish are precisely the points where all polynomials in I vanish and all polynomials in J vanish. Hence .
The product IJ of two ideals I and J is generated by all products of one element from I and one from J. Hence, an element of IJ is on the form with all the coming from I and all the coming from J. Suppose p is in V(IJ). Then for all f in I and all g in J, f(p)g(p)=0. If for all f in I, f(p)=0, then p is in V(I). Otherwise we can find some such that . But then since for all g in J, we must have for all g in J. Hence p is in V(J).
On the other hand, if p is in then specifically either p is in V(I) or it is in V(J). Either way, for all f in I and all g in J, f(p)g(p)=0 since one of the factors certainly vanishes. Hence, p is in V(IJ).
We also do have that . Which of the ideal operations works best depends a bit on what we would like to do with it.
We can use the above discussion to actually compute things geometrically. The intersection of the circle with a line is given by the sum of the ideals – so we need to consider the ideal in . We could try to bash this out by just wrangling the corresponding equations, or we could systematize the wrangling. Systematizing it leads, basically, to the theory of GrŲbner bases – and this is something I tend to avoid doing by hand if I can. Not doing it by hand looks like this:
$ Singular SINGULAR / A Computer Algebra System for Polynomial Computations / version 3-0-2 0< by: G.-M. Greuel, G. Pfister, H. Schoenemann \ July 2006 FB Mathematik der Universitaet, D-67653 Kaiserslautern \ > ring R=0,(x,y),dp; > ideal I=x^2+y^2-1,2*x-y; > std(I); _=2x-y _=5y2-4
So one good set of generators for the ideal would be . This, in turn corresponds to the conditions and . So we get two points – just as expected.
The equation wrangling, on the other hand, begins with recognizing that if both a and b are 0, then V(ax+by+c) is either empty or the entire plane. Both are rather boring, and easy to handle. So one of a and b is non-zero. Let’s say that a isn’t 0. Then we can rewrite ax+by+c=0 to . This can be inserted into the quadratic equation to form the new equation , which in turn expands to
which, in turn, we can easily solve using normal solution techniques for quadratic equations. To get rid of fractions, we’ll multiply the entire equation by yielding the new equation
Now, the discriminant
carries information about the nature of solutions. If , then the line V(ax+by+c) is a tangent to the circle, if , then the intersections are all complex and if , then the intersections are real. In both the non-tangent cases, the circle intersects the line exactly twice.