Del Pezzo Surfaces \\\\ A Note

Del Pezzo Surfaces A Note Johar M. Ashfaque In this note, we consider del Pezzo surfaces. The simplest ones are P1 × P1 (= F0 e.g. a Hirzebruch surface) and dP0 = P2 . Note. A Hirzebruch surface is a P1 fibration over a P1 ; the general type is classified by an integer index n, and denoted by Fn . There are eight more del Pezzo surfaces dPn constructed from an operation known as blow up of P2 at generic points. To blow-up a surface (manifold) at a point, we remove the point and replace it with a line gluing it in such a way so that we still get a manifold. The points of this line correspond to different directions from the marked point on the plane. Del Pezzo surfaces are obtained by applying the blow-up operation to up to eight points on the plane. A dPn is generated by the hyperplane divisor H from P2 and the exceptional divisors E1,...,8 with intersection numbers H · H = 1, H · Ei = 0, Ei · Ej = −δij . The canonical divisor and the first Chern class c1 (dP n) is given by KS = −c1 (dP n) = −3H +

n X

Ei .

i=1

The effective class C of a curve can be written as a sum of the generators Ci as X C= ni C i i

for ni > 0. The characteristic property of a del Pezzo surface is that c1 is ample that is equivalent to saying it has positive intersection with every effective curve. This in particular implies that K must have positive self-intersection KS · KS = 9 − n which gives the restriction n ≤ 8. A K¨ ahler class can be defined as follows ω = AH −

n X

ai Ei .

i=1

For a line bundle L on a del Pezzo surface with c1 (L) =

n X

mi Ei

i=1

where mi are integers, the condition ω · c1 (L) = 0 implies X ai mi = 0 i

while for sufficiently large A and for any divisor D, the intersection is positive ω · D > 0. To see the connection of dPn with exceptional Lie algebras, define the generators for n ≥ 3 a1 = E1 − E2 , ..., an−1 = En−1 − En , an = H − E1 − E2 − E3

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Using the dot product for the generators, we find

ai · aj = 2δij − δi,j+1 − δj,i+1

   2, i = j = −1, i = j + 1   −1, j = i + 1

The intersection product of ai is identical to minus the Cartan matrix for the dot product of the simple roots of the corresponding algebra En . In the particular case of dP2 there is only one generator E1 − E2 which is identified as a root of SU (2).

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