Algebraic geometry

Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. It can be seen as the study of solution sets of systems of algebraic equations. When there is more than one variable, geometric considerations enter, and are important to understand the phenomenon. One can say that the subject starts where equation solving leaves off, and it becomes at least as important to understand the totality of solutions of a system of equations as to find some solution; this does lead into some of the deepest waters in the whole of mathematics, both conceptually and in terms of technique.

Projective space

Consider the variety V(y=x2). If we draw it, we get a parabola. As x increases, the slope of the line from the origin to the point (x,x2) becomes larger and larger. As x decreases, the slope of the same line becomes smaller and smaller.

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Compare this to the variety V(y=x3). This is a cubic equation. As x increases, the slope of the line from the origin to the point (x,x3) becomes larger and larger just as before. But unlike before, as x decreases, the slope of the same line again becomes larger and larger. So the behavior "at infinity" of V(y=x3) is different from the behavior "at infinity" of V(y=x2). It is, however, difficult to make the concept of "at infinity" meaningful, if we restrict to working in affine space.

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The remedy to this is to work in projective space. Projective space has properties analogous to those of a compact Hausdorff space. Among other things, it lets us make precise the notion of "at infinity" by including extra points. The behavior of a variety at those extra points then gives us more information about it. As it turns out, V(y=x3) has a singularity at one of those extra points, but V(y=x2) is smooth.

Related Topics:
Projective space - Compact - Hausdorff space - Singularity

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While projective geometry was originally established on a synthetic foundation, the use of homogeneous coordinates allowed the introduction of algebraic techniques. Furthermore, the introduction of projective techniques made many theorems in algebraic geometry simpler and sharper: For example, Bézout's theorem on the number of intersection points between two varieties can be stated in its sharpest form only in projective space. For this reason, projective space plays a fundamental role in algebraic geometry.

Related Topics:
Projective geometry - Synthetic - Homogeneous coordinates - Bézout's theorem

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