Complex Algebraic Surfaces Class 14
نویسنده
چکیده
Useful proposition. Consider the blow-up of P at n general points, giving exceptional divisors E1, : : : , En. Then the intersection ring on P is given by Z[H;E1; : : : ; En]=H 2 = 1; HEi = 0; Ei Ej = 0; E 2 i = 1: We can understand divisors and sections of divisors in terms of divisors on P with certain multiplicities at the Ej . More precisely: the vector space of sections of aH b1E1 bnEn is naturally isomorphic to the vector space of degree a polynomials in P vanishing with multiplicity at least bi on Ei. We also proved the following. Suppose S is a surface, and KS gives a map to projective space. Then ( 1)-curves map isomorphically onto lines. Conversely, if S is a surface, and KS gives a map to projective space, then any curve mapping isomorphically onto a line is a ( 1)-curve.
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