By Arnaud Beauville
ISBN-10: 0521495105
ISBN-13: 9780521495103
The category of algebraic surfaces is an problematic and interesting department of arithmetic, built over greater than a century and nonetheless an lively sector of analysis this present day. during this e-book, Professor Beauville provides a lucid and concise account of the topic, expressed easily within the language of recent topology and sheaf idea, and available to any budding geometer. A bankruptcy on initial fabric guarantees that this quantity is self-contained whereas the workouts be successful either in giving the flavour of the classical topic, and in equipping the reader with the options wanted for examine. The publication is geared toward graduate scholars in geometry and topology.
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Sample text
This follows from the next lemma (a form of the adjunction formula). 11 Let S C 1pr+2 be a surface that is the complete intersection of hypersurfaces Hl, ... , H, of degrees d1,.. , d,. respectively. Then Os(Ks)-Os(F, di-r-3). Proof Let I be the ideal defining S C P''+2. Since the equations of the Hi generate I, there is a surjection Ors+2(-di) ® ® Opr+s(-d,) -+ I, IV: Rational Surfaces 47 and hence a surjection (by restricting to S) u :Os(-di)®... ® Os(-d,) --+I/I2 Both sides are locally free of rank r, and so u is an isomorphism.
Although blow-ups appear in their construction, they are not given prominence; the structure theorem for birational maps seems only to have been observed much later by Zariski ([Z1, p. 538], 1944). g. [Z2]); we have followed his ideas, as they are described in [Shl]. 20 (1) Let C be an irreducible curve on a surface S. Show that there is a morphism S -+ S consisting of a finite number of blow-ups such that the strict transform of C in S is smooth. ) (2) Let C be an irreducible curve on S, p E C and C the strict transform of C on the blow-up with centre p.
Ai,k_i of H°(S, Os(H+iE)) whose restrictions to E form a basis of H°(E,OE(k - i)). Then {SkSo,... )s k sn, sk-1a1,0, ... )sk-1 al,k-1, ... , sak_1,1i ak,o} is a basis of H°(S,Os(H')). Let 0 : S --+ P''` denote the rational map defined by the corresponding linear system. Since the map defined by ... , sn } is an embedding, the restriction of 0 to S - E is an embedding. Since ak,o induces a non-zero constant function on E, we see that 0 is defined everywhere and contracts E to the point p = (0, ...
Complex Algebraic Surfaces by Arnaud Beauville
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