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created: 2024-04-28T13:11
updated: 2024-04-28T13:11
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# Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville

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## Pages

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=7|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 7]]

> ruled surfaces, that is, surfaces birational to lP x C.

> their minimal models are Pl-bundles over a base curve C,

> (the Veronese surface, del Pezzo surfaces,

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=8|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 8]]

> p9 = 0 and q > 2 is ruled;

> Thus a surface is ruled if and only if P12 = 0 (Enriques' theorem).

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=11|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 11]]

> If f is surjective,

> f * Ox (D) = Os (f * (D))

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=12|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 12]]

> f. f * D = dD for all divisors D on S

> m.,(C fl C') = 1 if and only if f and g generate the maximal ideal ,;, i.e. form a system of local coordinates in a neighbourhood of x : C and C' are then said to be transverse at x.

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=13|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 13]]

> deg(Llc)

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=14|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 14]]

> g*D.g*D' = d(D.D').

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=15|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 15]]

### [[Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville.pdf#page=16|Complex Algebraic Surfaces, 2nd ed_ - Arnaud Beauville, page 16]]

> T = H1(S, Os)/H1(S, Z)

> that H'(S,Z) is a lattice in H'(S, Os), so T has a natural structure of complex torus);