By Arnaud Beauville

ISBN-10: 0521495105

ISBN-13: 9780521495103

The class of algebraic surfaces is an problematic and engaging department of arithmetic, built over greater than a century and nonetheless an lively zone of study at the present time. during this publication, Professor Beauville offers a lucid and concise account of the topic, expressed easily within the language of contemporary topology and sheaf thought, and obtainable to any budding geometer. A bankruptcy on initial fabric guarantees that this quantity is self-contained whereas the routines prevail either in giving the flavour of the classical topic, and in equipping the reader with the recommendations wanted for examine. The e-book is geared toward graduate scholars in geometry and topology.

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Let Lit = (pi, pj) for i # j. (a) We want to show that this linear system separates points on P6. So let x, y E P6 with x # y. Choose i with pi $ e(x), e(y) and fl - = {x} for pk 0 {pi, pj, e(x)}. Hence x 0 Q. Then y E Q for at most one value of j. On the other, hand, y E Lij for at most one j; thus there is a j such that the cubic Q U Lii passes through x but not y. Hence the morphism j : P6 --+ IEn3 is injective. (b) Let x E P2 - {PI, ... , P6}. The cubits Qi U (pi, x) do not all have the same tangent at x, so that j is an immersion at x.

18 Let S = ]PC(E) be a geometrically ruled surface over C, p : S --* C the structure map. Write h for the class of the sheaf Os(l) in PicS (or in H2(S,7L)). Then (i) Pic S = p* Pic C ®7Lh. (ii) H2(S,7L) = 7Lh ® 7Lf, where f is the class of a fibre. (iii) h2 = deg(E). (iv) [K] = -2h + (deg(E) + 2g(C) - 2) f in H2(S, Z). Proof First we prove (i). F = 0; so it is enough to prove that D is the pull-back of a divisor on C. K - 2n, and h°(K - Dn) = 0 whenever n is sufficiently large. , so that the system ID,, I is non-empty for sufficiently large n.

Show that V is the set of conics of rank 1 in P (use (5)(a)). Deduce that V is the intersection of 5 quadrics in Q. Show that the union of the bisecants of V is the set X of singular conics in P. Deduce that X is a cubic hypersurface in (Q whose singular locus is V. (6) Let f : 1112 -> I(1s be the morphism corresponding to a 3-dimensional base-point free linear system of conics. Let R be the dual pencil of conics in ](D2; suppose that R contains 3 distinct singular conics, each of rank 2. These correspond to 3 pairs of points (pi, p;), i = 1, 2, 3.

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