1. Warming up to Enumerative Geometry 9 any of its factors. Sometimes we will want to consider multiplicities and then we have to put extra powers back in. In particular we can state an easy converse of Theorem 1.2 as say- ing that any set Z P1 consisting of d points (including multiplic- ity) is the zero locus of a degree d homogeneous polynomial f(x0, x1), unique up to a scalar multiple. If we have points (ai, bi) P1 with multiplicities ri and i ri = d, we can take f to be the degree d polynomial f(x0, x1) = i (bix0 aix1)ri. Just as for P1, we have a notion of the dehomogenization of F (x0, . . . , xn): (5) f(x1, . . . , xn) = F (ψ0(x1, . . . , xn)) = F (1, x1, . . . , xn) and a notion of the homogenization of f(x1, . . . , xn): (6) F (x0, . . . , xn) = xdf 0 x1 x0 , . . . , xn x0 , where d is the degree of f(x1, . . . , xn), the maximum total degree of any term in f. Compare with [Hulek, Sections 2.1, 2.2], a text containing an in- troduction to algebraic geometry with minimal prerequisites. [Reid] and [Fulton3] are other algebraic geometry texts readily accessible to undergraduates. For graduate level algebraic geometry, see, e.g., [Miranda] for complex algebraic geometry in dimension 1, [GH] for complex algebraic geometry more generally, or see either [Harris] or [Hartshorne] for algebraic geometry over arbitrary fields. By a line in P2, we mean the hypersurface defined by a homoge- neous polynomial of degree 1 in (x0, x1, x2) (degree 1 hypersurfaces in Pn are more generally called hyperplanes). In other words, the poly- nomial ax + by = c in C2 can be homogenized to get ax1 + bx2 = cx0, and there is a similar homogenization for dx+ey = f (note that (x, y) has been replaced by (x1, x2) here). Now lines that were parallel in the plane meet in P2! Consider for instance the lines x + y = 1 and x + y = 2. Their homogenizations are x1 + x2 = x0 and x1 + x2 = 2x0. Subtracting these equations,
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