By Arnaud Beauville

The category of algebraic surfaces is an complex and interesting department of arithmetic, built over greater than a century and nonetheless an lively zone of study at the present time. during this ebook, Professor Beauville supplies a lucid and concise account of the topic, expressed easily within the language of recent 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 be successful either in giving the flavour of the classical topic, and in equipping the reader with the options wanted for learn. The ebook is geared toward graduate scholars in geometry and topology.

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Checking the numbers of the curves is immediate. ). These configurations were studied intensively by the classical geometers - cf. Exercises 12, 13, 14. 13 Let S C PE be a smooth cubic surface. e. is isomorphic to PP'2 with 6 points blown up). First we prove two lemmas. 14 S contains a line. Proof Let P = JOr3(3)l be the projective space of cubics of P1, and let G4 denote the Grassmannian of lines in P3 (since a line in P3 is given by 6 Plucker coordinates, subject to a single quadratic relation, G4 is naturally a smooth quadric in Il5, and is 4-dimensional).

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. Now let IV: Rational Surfaces 46 X E El; the conics Q23 and Q24 intersect at x with multiplicity 2. Then the cubics Q23 U L23 and Q24 U L24 have different tangents at x, which completes the proof that j is an embedding. 2(4), deg(j(P,)) = 9-r, so that j(P,) = Sd is a smooth surface of degree d in Pd, with d = 9 - r. In particular, S3 is a cubic surface in P3 and S4 is of degree 4 in Pd.

Thus S4 lies in two distinct quadrics Ql and Q2, which must be irreducible; Ql fl Q2 is then a surface of degree 4 containing S4, and so equal to it. 10 (1) Note that the linear system of cubics through pl, ... , p, is in fact the complete `anticanonical system' I - K I on P,. One can show that together with P1 x P1 embedded in Id, the del Pezzo surfaces are the only ones embedded in P' by their complete anticanonical system (Chapter V, Exercise 2). (2) Cubics and complete intersections of two quadrics are the only complete intersection surfaces embedded by their anticanonical system.

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