By David Eisenbud and Joseph Harris

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10 the intersection Γf ∩ (Λ × Φ) is generically transverse. Finally, Γf ∩ (Λ × Φ) is the zero locus, in Λ, of r − i general linear combinations of the polynomials F0 , . . , Fr ; by Bertini’s Theorem (Hartshorne [1977] Theorem ****) the corresponding hypersurfaces will intersect transversely, and by B´ezout’s Theorem the intersection will consist of dr−i points. 25. If f : P r → P s is a regular map given by polynomials of degree d on P r , the class γf of the graph of f is given by r di αi β s−i ∈ As (P r × P s ).

60. Now let P 14 be the space of quartic curves in P 2 . (a) Let Σ ⊂ P 14 be the closure of the space of reducible quartics. What are the irreducible components of Σ, and what are their dimensions and degrees? (b) Find the dimension and degree of the locus of totally reducible quartics (that is, quartic polynomials that factor as a product of four linear forms). 61. Use the Poincar´e-Hopf Theorem to compute the topological Euler characteristic of a smooth variety Y = X1 ∩ X2 ⊂ P n where Xi is a hypersurface of degree di .

Ys ] → [. . , Xi Yj , . . ]. It is easy to check that the map σr,s is an immersion. If V and W are vector spaces of dimensions r + 1 and s + 1 we may write σr,s without bases by the formula σr,s : PV × PW → P(V ⊗ W ) (v, w) → v ⊗ w. For example, the map σ1,1 is defined by the four forms a = X0 Y0 , b = X0 Y1 , c = X1 Y0 , d = X1 Y1 , and these satisfy the equation ac − bd = 0, the Segre variety Σ1,1 is the nonsingular quadric in P 3 . To compute the degree of Σr,l in general, we proceed as with the Veronese varieties.