By David Eisenbud and Joseph Harris

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**Example text**

A cycle Z = i ni Yi is called effective if the coefficients ni are all nonnegative. A divisor is a cycle whose components all have codimension 1. Note that Z∗ (X) = Z∗ (Xred ); that is, Z∗ (X) is insensitive to whatever non-reduced structure X may have. We define an effective cycle Z associated with any closed subscheme Y ⊂ X by summing the irreducible components of Y , each with a multiplicity defined using the following algebraic idea: If A is any commutative ring then we say that an A-module M has finite length if it has a finite composition series; that is, a sequence of submodules M = M0 M1 · · · Ml = 0 such that each factor module Mi /Mi+1 is a simple A-module.

C) Show that the exceptional divisor E of the blow-up is isomorphic to P 1 × P 1 . Now let A, B ⊂ E be the classes of the fibers of the two projections and let a, b ∈ A2 (X) be their classes. Find the degrees of the products ea, eb. (d) Using the above, give a complete description of the Chow ring of X. 49. Let L and C ⊂ P 2 be a line and a conic intersecting transversely at two points p, q ∈ P 2 ; let [L + C] be the corresponding point of Γ. Show that Γ is smooth at [L + C], with tangent space T [L+C] Γ = {homogeneous cubic polynomials F : F (p) = F (q) = 0}.

Let Fi ([X0 , . . , Xr ], [Y0 , . . , Ys ]) be general bihomogeneous polynomials of bidegree (d, e), for i = 0, . . , r + s + 1. For how many X = [X0 , . . , Xr ] and Y = [Y0 , . . , Ys ] do we have [F0 (X, Y ), . . , Fr (X, Y )] ∼ [X0 , . . , Xr ] and [Fr+1 (X, Y ), . . , Fr+s+1 (X, Y )] ∼ [Y0 , . . , Ys ]? 44. Consider the locus Φ ⊂ (P 2 )4 of fourtuples of collinear points. 41 on the locus Ψ ⊂ (P 2 )3 of triples of collinear points, and considering the intersection of the loci of fourtuples (p1 , p2 , p3 , p4 ) with p1 , p2 , p3 collinear and with p1 , p2 , p4 collinear.