By C. R. F. Maunder

Thorough, smooth remedy, primarily from a homotopy theoretic standpoint. issues comprise homotopy and simplicial complexes, the elemental workforce, homology idea, homotopy conception, homotopy teams and CW-Complexes and different themes. each one bankruptcy comprises routines and recommendations for extra interpreting. 1980 corrected variation.

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**Example text**

Upper bounds and submaximal curves. 9]. For 0-dimensional reduced subschemes we have the following result. 1 (Upper bounds). Let X be a smooth projective variety of dimension n and L a nef line bundle on X. Let x1 , . . , xr be r distinct points on X, then n n L . ε(X, L; x1 , . . , xr ) r In particular for a single point x we always have √ n ε(X, L; x) Ln . Proof. Let f : Y −→ X be the blowup x1 , . . , xr . Then the exceptional divisor E = E1 + · · · + Er is the sum of disjoint exceptional divisors over each of the points.

1) h−1 ([σ0 , . . , σr1 ], [τ0 , . . , τr2 ]) ⊂ g −1 (Y ). Let N1 denote the line of P(H 0 (OPn1 (1)⊕(r1 +1) ) ∼ = P(V1 1 ) passing through the points [s1 , . . , sr1 , 0] and [0, s1 , . . , sr1 ], and N2 the line of P(H 0 (OPn2 (1)⊕(r2 +1) ) ∼ = ⊕(r +1) ˘ LUCIAN BADESCU 16 6 ⊕(r +1) P(V2 2 ) passing through the points [t1 , . . , tr2 , 0] and [0, t1 , . . , tr2 ]. 2, h−1 (N1 × N2 ) is also G3 in Z (because h is proper and surjective). On the other hand, since every point of the line N1 is of the form [λs1 , λs2 + µs1 , .

Tr2 , 0] and [0, t1 , . . , tr2 ]. 2, h−1 (N1 × N2 ) is also G3 in Z (because h is proper and surjective). On the other hand, since every point of the line N1 is of the form [λs1 , λs2 + µs1 , . . 1) we get the inclusion h−1 (N1 × N2 ) ⊂ g −1 (Y ), and therefore the natural maps of k-algebras K(Z) → K(Z/g−1 (Y ) ) → K(Z/h−1 (N1 ×N2 ) ), whose composition is an isomorphism because h−1 (N1 × N2 ) is G3 in Z. We claim that this implies that the ﬁrst map is also an isomorphism. To see this, it will be enough to check that K(Z/g−1 (Y ) ) is a ﬁeld.