Del Pezzo surface
A Del Pezzo surface is a smooth projective algebraic surface over an algebraically closed field with an ample anticanonical divisor.[1] Equivalently, it is a smooth surface of degree d embedded in projective space \mathbb{P}^d for d \geq 3.[2] Such surfaces are classified by their degree d = (-K_X)^2, which ranges from 1 to 9, where K_X denotes the canonical divisor.[2] For degrees 3 through 8, a Del Pezzo surface of degree d is isomorphic to the blow-up of the projective plane \mathbb{P}^2 at r = 9 - d points in general position—no three collinear and no six on a conic.[1] The cases of degree 9 correspond to \mathbb{P}^2 itself, while degree 8 includes the quadric surface \mathbb{P}^1 \times \mathbb{P}^1, and degree 1 or 2 surfaces have anticanonical models as hypersurfaces in weighted projective spaces.[2][3] Del Pezzo surfaces exhibit rich geometric structures, including a finite number of exceptional curves (rational curves of self-intersection -1) whose configuration corresponds to root systems of Lie algebras, such as E_6 for the cubic surface of degree 3 with its 27 lines.[2] The anticanonical linear system provides an embedding into projective space, and these surfaces play a central role in birational geometry, enumerative problems, and connections to representation theory.[2] Named after the Italian mathematician Pasquale del Pezzo, who studied cubic surfaces in the 1880s, they were further developed in the 20th century through links to Weyl groups and homogeneous spaces.[4]Introduction
Definition
A Del Pezzo surface over a field k is defined as a smooth projective surface X such that the anticanonical divisor -K_X is ample. This condition ensures that X admits an anticanonical embedding into projective space via the complete linear system | -K_X |, making it a Fano variety of dimension 2.[5] Del Pezzo surfaces are not ruled unless isomorphic to \mathbb{P}^1 \times \mathbb{P}^1.[6] The degree d of a Del Pezzo surface is given by the self-intersection d = (-K_X)^2 = K_X^2, which takes integer values from 1 to 9. Over an algebraically closed field, every Del Pezzo surface is rational. In particular, they can be realized as blow-ups of \mathbb{P}^2 at up to 8 points.[5]Historical Context
The study of Del Pezzo surfaces originated in the late 19th century with the work of the Italian mathematician Pasquale del Pezzo, who examined non-ruled, non-degenerate surfaces of degree n embedded in projective space \mathbb{P}^n. In his 1887 paper, del Pezzo analyzed these surfaces, noting their projectivity from general points and their connections to lower-degree cases like cubic surfaces, including the configuration of lines on such cubics.[7] His contributions laid the foundation for understanding these surfaces as rational varieties with specific embedding properties, influencing subsequent classifications in algebraic geometry.[8] In the early 20th century, the concept evolved through connections to broader classes of varieties. Gino Fano, in his 1928 address at the International Congress of Mathematicians, explored three-dimensional algebraic varieties with all genera zero, pioneering the study of what would later be termed Fano varieties—projective varieties with ample anticanonical divisor.[9] Del Pezzo surfaces, as two-dimensional Fano varieties, fit naturally into this framework, with Fano's work in the 1930s further emphasizing varieties where the anticanonical bundle plays a central role in birational properties.[10] Advancements in the mid-20th century formalized the modern perspective on these surfaces. In the 1960s, the Proceedings of the Steklov Institute on algebraic surfaces, edited by I.R. Shafarevich, integrated sheaf theory and cohomology to describe Del Pezzo surfaces via the ampleness of the anticanonical sheaf, distinguishing them from del Pezzo's original very ample condition and linking them to rational surfaces whose Picard lattices have an indefinite intersection form, with the orthogonal complement to the canonical class being negative definite and corresponding to root systems of Lie algebras.[5] This era solidified their role in the classification of algebraic surfaces. Post-1980s developments in birational geometry, particularly through the minimal model program introduced by Shigefumi Mori, elevated the importance of Del Pezzo surfaces as building blocks in the study of Fano varieties and rational maps. Mori's bend-and-break technique and contraction theorems highlighted their minimal nature and connections to higher-dimensional Fanos, integrating them into contemporary programs for variety classification.Properties
Anticanonical Divisor and Degree
A Del Pezzo surface is characterized by its anticanonical divisor -K being ample, which implies that the surface is a Fano variety and stands in contrast to surfaces of general type where the canonical divisor K is ample.[11] The ampleness of -K ensures that the surface has positive intersection numbers with effective divisors and supports a rich linear system structure essential for its embedding properties.[11] The degree d of a Del Pezzo surface X is defined as d = K_X^2 = (-K_X)^2, where K_X denotes the canonical divisor; this self-intersection number satisfies $1 \leq d \leq 9, with equality to 9 holding if and only if X \cong \mathbb{P}^2.[11] A fundamental theorem states that if a smooth projective surface admits an ample divisor D with D^2 > 9, then the surface is ruled; in particular, for Del Pezzo surfaces where D = -K, this bound restricts d \leq 9.[11] Moreover, when d = 9, the complete linear system | -K | is given by \mathcal{O}_{\mathbb{P}^2}(3), which realizes \mathbb{P}^2 as the Veronese surface of degree 9 in \mathbb{P}^9.[11] In intersection theory, the anticanonical divisor -K on a Del Pezzo surface of degree d \geq 3 is very ample, providing a closed embedding of the surface into \mathbb{P}^d as a surface of degree d.[11] This embedding highlights the surface's minimal model properties within projective space.Lines and Exceptional Curves
On a Del Pezzo surface of degree d, an exceptional curve, also known as a (-1)-curve, is defined as an irreducible rational curve C with self-intersection number C^2 = -1 and intersection number (-K_X \cdot C) = 1, where K_X is the canonical divisor.[12] These curves are isomorphic to \mathbb{P}^1 by the adjunction formula, as their arithmetic genus is zero.[12] The number of exceptional curves on a Del Pezzo surface varies with the degree d: there are 27 such curves for d=[3](/page/3), 56 for d=2, and 240 for d=1.[12] Their classes in the Picard group form a root system of type E_6 for d=3, E_7 for d=2, and E_8 for d=1, reflecting the combinatorial structure inherited from the blow-up construction at $9-d points in \mathbb{P}^2.[2] Distinct exceptional curves are either disjoint (skew) or intersect transversely at exactly one point, with intersection number 0 or 1.[13] Contracting an exceptional curve via the blow-down map yields a birational morphism to another Del Pezzo surface of degree d+1, preserving the Del Pezzo property until reaching \mathbb{P}^2 for d=9.[12] The classes of these exceptional curves generate the effective cone of curves on the surface, and their configuration, governed by the action of the corresponding Weyl group, determines the isomorphism class of the surface in characteristic zero.[2]Classification
Over Algebraically Closed Fields
Over an algebraically closed field k, del Pezzo surfaces are classified up to isomorphism by their degree d = (-K_X)^2, where $1 \leq d \leq 9. Every such surface is rational and minimal with respect to contractions of (-1)-curves. Specifically, for d = 9, the surface is isomorphic to \mathbb{P}^2_k. For $1 \leq d \leq 8, it is either the blow-up of \mathbb{P}^2_k at r = 9 - d points in general position or, in the case d = 8, the quadric \mathbb{P}^1_k \times \mathbb{P}^1_k. The points are in general position if no three are collinear, no six lie on a conic, and—for r = 8—no eight lie on a cubic curve.[12] This classification implies uniqueness of the minimal model up to isomorphism for each degree d = 3, \dots, 9, except for d = 8, where there are precisely two non-isomorphic types: the blow-up of \mathbb{P}^2_k at one point and \mathbb{P}^1_k \times \mathbb{P}^1_k. For d = 1 and d = 2, the surfaces are also unique up to isomorphism as the blow-ups of \mathbb{P}^2_k at eight and seven points in general position, respectively. These results hold over algebraically closed fields of characteristic zero, with analogous statements in positive characteristic under mild assumptions on the characteristic relative to the degree.[12] The anticanonical divisor -K_X embeds del Pezzo surfaces of degree d \geq 3 as smooth hypersurfaces of degree d in \mathbb{P}^d_k. In particular, for d = 3, this is a smooth cubic surface in \mathbb{P}^3_k. For d = 2, the anticanonical model is a smooth quartic hypersurface in the weighted projective space \mathbb{P}_k(1,1,1,2), which realizes the surface as a double cover of \mathbb{P}^2_k branched over a smooth quartic curve. For d = 1, it is a smooth sextic hypersurface in \mathbb{P}_k(1,1,2,3). In these low-degree cases, -K_X is ample but not very ample, leading to models in weighted projective spaces rather than ordinary projective space.[12] The following table enumerates the isomorphism classes of del Pezzo surfaces over an algebraically closed field k by degree:| Degree d | Isomorphism class |
|---|---|
| 9 | \mathbb{P}^2_k |
| 8 | \mathrm{Bl}_1 \mathbb{P}^2_k or \mathbb{P}^1_k \times \mathbb{P}^1_k |
| 7 | \mathrm{Bl}_2 \mathbb{P}^2_k |
| 6 | \mathrm{Bl}_3 \mathbb{P}^2_k |
| 5 | \mathrm{Bl}_4 \mathbb{P}^2_k |
| 4 | \mathrm{Bl}_5 \mathbb{P}^2_k |
| 3 | \mathrm{Bl}_6 \mathbb{P}^2_k |
| 2 | \mathrm{Bl}_7 \mathbb{P}^2_k |
| 1 | \mathrm{Bl}_8 \mathbb{P}^2_k |