DelPezzoFlächen

DelPezzoFlächen, also known as Del Pezzo surfaces, are a class of algebraic surfaces in algebraic geometry. They are defined as the set of points satisfying a certain homogeneous polynomial equation. Specifically, a Del Pezzo surface is a smooth, projective algebraic surface obtained by blowing up the projective plane $P^2$ at a finite number of points, say $n$ points, such that the exceptional curves of the blow-up are disjoint. The maximum number of points that can be blown up while maintaining the disjointness of exceptional curves is 8.

The classification of Del Pezzo surfaces depends on the number of points blown up. A Del Pezzo surface of degree $d$ is a surface that can be realized as a quartic hypersurface in $P^3$ with exactly $d$ nodes. Alternatively, it can be viewed as the blow-up of $P^2$ at $9-d$ points. The case $d=1$ corresponds to the blow-up of $P^2$ at 8 points, which is the most general Del Pezzo surface. The case $d=2$ is the blow-up of $P^2$ at 7 points, and so on. The case $d=9$ corresponds to $P^2$ itself, which is considered a Del Pezzo surface of degree 9.

Del Pezzo surfaces possess rich geometric properties. They are rational surfaces, meaning they are birationally equivalent to the projective plane. They also have a large number of rational curves on them. The Picard group of a Del Pezzo surface is related to the root system of type $E_8$. These surfaces play a significant role in various areas of algebraic geometry, including birational geometry and the study of moduli spaces. Their properties make them important examples for understanding the behavior of algebraic varieties.