Del Pezzo surface
Encyclopedia
In mathematics
, a del Pezzo surface or Fano surface is a two-dimensional Fano variety
, in other words a non-singular projective algebraic surface
with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.
They are named for Pasquale del Pezzo
who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space , which are the del Pezzo surfaces of degree at least 3.
The degree d of a del Pezzo surface X is by definition the self intersection number (K, K) of its canonical class K.
A −1-curve is a rational curve with self intersection number −1. For d > 2, the image of such a curve in projective space under the anti-canonical embedding is a line.
Any curve on a del Pezzo surface has self intersection number at least −1. The number of curves with self intersection number −1 is finite and depends only on the degree (unless the degree is 8).
The blowdown of any −1-curve on a del Pezzo surface is a del Pezzo surface of degree 1 more.
The blowup of any point on a del Pezzo surface is a del Pezzo surface of degree 1 less, provided that the point does not lie on a −1-curve and the degree is greater than 1.
Del Pezzo proved that a del Pezzo surface has degree d at most 9. Over an algebraically closed field, every del Pezzo surface
is either a product of two projective lines (with d=8), or the blow-up of a projective plane in 9 − d points with no three collinear, no six on a conic, and no eight of them on a cubic having a node at one of them.
Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.
The Picard group of a del Pezzo surface of degree d is the odd unimodular lattice
I1,9−d, except when the surface is a product of 2 lines when the Picard group is the even unimodular lattice II1,1.When it is an odd lattice, the canonical element is (3, 1, 1, 1, ....), and the exceptional curves are represented by permutations of all but the first coordinate of the following vectors:
Degree 2: they have 56 −1-curves corresponding to the minuscule vectors of the dual of the E7 lattice. They form a 6-dimensional family. The anticanonical divisor is not very ample, and its linear system defines a map from the del Pezzo surface to the projective plane, branched over a quartic plane curve
. This map is generically 2 to 1, so this surface is sometimes called a del Pezzo double plane. The 56 lines of the del Pezzo surface map in pairs to the 28 bitangents of a quartic.
Degree 3: these are essentially cubic surface
s in P3; the cubic surface is the image of the anticanonical embedding. They have 27 −1-curves corresponding to the minuscule vectors of one coset in the dual of the E6 lattice, which map to the 27 lines of the cubic surface. They form a 4-dimensional family.
Degree 4: these are essentially Segre surface
s in P4, given by the intersection of two quadrics. They have 16 −1-curves. They form a 2-dimensional family.
Degree 5: they have 10 −1-curves corresponding to the minuscule vectors of one coset in the dual of the A4 lattice. There is up to isomorphism only one such surface, given by blowing up the projective plane in 4 points with no 3 on a line.
Degree 6: they have 6 −1-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 3 points not on a line. The root system is A2 × A1
Degree 7: they have 3 −1-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 2 distinct points.
Degree 8: they have 2 isomorphism types. One is a Hirzebruch surface
given by the blow up of the projective plane at one point, which has 1 −1-curves. The other is the product of two projective lines, which is the only del Pezzo surface that cannot be obtained by starting with the projective plane and blowing up points. Its Picard group is the even 2-dimensional unimodular indefinite lattice II1,1, and it contains no −1-curves.
Degree 9: The only degree 9 del Pezzo surface is the P2. Its anticanonical embedding is the degree 3 Veronese embedding into P9 using the linear system of cubics.
The blowdown of any −1-curve on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 more.
The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2-curve and the degree is greater than 1.
Any curve on a weak del Pezzo surface has self intersection number at least −2. The number of curves with self intersection number −2 is at most 9−d, and the number of curves with self intersection number −1 is finite.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a del Pezzo surface or Fano surface is a two-dimensional Fano variety
Fano variety
In algebraic geometry, a Fano variety, introduced by , is a non-singular complete variety whose anticanonical bundle is ample.Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces....
, in other words a non-singular projective algebraic surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general type, which have ample canonical class.
They are named for Pasquale del Pezzo
Pasquale del Pezzo
Pasquale del Pezzo, Duke of Cajanello and Marquis of Campodisola , was a Neapolitan mathematician.He was born in Berlin on 2 May 1859. He died in Naples on 20 June 1936...
who studied the surfaces with the more restrictive condition that they have a very ample anticanonical divisor class, or in his language the surfaces with a degree n embedding in n-dimensional projective space , which are the del Pezzo surfaces of degree at least 3.
Classification
A del Pezzo surface is a complete non-singular surface with ample anticanonical bundle. There are some variations of this definition that are sometimes used. Sometimes del Pezzo surfaces are allowed to have singularities. They were originally assumed to be embedded in projective space by the anticanonical embedding, which restricts the degree to be at least 3.The degree d of a del Pezzo surface X is by definition the self intersection number (K, K) of its canonical class K.
A −1-curve is a rational curve with self intersection number −1. For d > 2, the image of such a curve in projective space under the anti-canonical embedding is a line.
Any curve on a del Pezzo surface has self intersection number at least −1. The number of curves with self intersection number −1 is finite and depends only on the degree (unless the degree is 8).
The blowdown of any −1-curve on a del Pezzo surface is a del Pezzo surface of degree 1 more.
The blowup of any point on a del Pezzo surface is a del Pezzo surface of degree 1 less, provided that the point does not lie on a −1-curve and the degree is greater than 1.
Del Pezzo proved that a del Pezzo surface has degree d at most 9. Over an algebraically closed field, every del Pezzo surface
is either a product of two projective lines (with d=8), or the blow-up of a projective plane in 9 − d points with no three collinear, no six on a conic, and no eight of them on a cubic having a node at one of them.
Conversely any blowup of the plane in points satisfying these conditions is a del Pezzo surface.
The Picard group of a del Pezzo surface of degree d is the odd unimodular lattice
Unimodular lattice
In mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
I1,9−d, except when the surface is a product of 2 lines when the Picard group is the even unimodular lattice II1,1.When it is an odd lattice, the canonical element is (3, 1, 1, 1, ....), and the exceptional curves are represented by permutations of all but the first coordinate of the following vectors:
- (0, −1, 0, 0, ....) the exceptional curves of the blown up points,
- (1, 1, 1, 0, 0, ...) lines through 2 points,
- (2, 1, 1, 1, 1, 1, 0, ...) conics through 5 points,
- (3, 2, 1, 1, 1, 1, 1, 1, 0, ...) cubics through 7 points with a double point at one of them,
- (4, 2, 2, 2, 1, 1, 1, 1, 1) quartics through 8 points with double points at three of them,
- (5, 2, 2, 2, 2, 2, 2, 1, 1) quintics through 8 points with double points at all but two of them,
- (6, 3, 2, 2, 2, 2, 2, 2, 2) sextics through 8 points with double points at all except a single point with multiplicity three.
Examples
Degree 1: they have 240 −1-curves corresponding to the roots of an E8 root system. They form an 8-dimensional family. The anticanonical divisor is not very ample. The linear system |−2K| defines a degree 2 map from the del Pezzo surface to a quadratic cone in P3, branched over a nonsingular genus 4 curve cut out by a cubic surface.Degree 2: they have 56 −1-curves corresponding to the minuscule vectors of the dual of the E7 lattice. They form a 6-dimensional family. The anticanonical divisor is not very ample, and its linear system defines a map from the del Pezzo surface to the projective plane, branched over a quartic plane curve
Quartic plane curve
A quartic plane curve is a plane curve of the fourth degree. It can be defined by a quartic equation:Ax^4+By^4+Cx^3y+Dx^2y^2+Exy^3+Fx^3+Gy^3+Hx^2y+Ixy^2+Jx^2+Ky^2+Lxy+Mx+Ny+P=0....
. This map is generically 2 to 1, so this surface is sometimes called a del Pezzo double plane. The 56 lines of the del Pezzo surface map in pairs to the 28 bitangents of a quartic.
Degree 3: these are essentially cubic surface
Cubic surface
A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
s in P3; the cubic surface is the image of the anticanonical embedding. They have 27 −1-curves corresponding to the minuscule vectors of one coset in the dual of the E6 lattice, which map to the 27 lines of the cubic surface. They form a 4-dimensional family.
Degree 4: these are essentially Segre surface
Segre surface
In algebraic geometry, a Segre surface, named for Beniamino Segre, is an intersection of two quadrics in 4-dimensional projective space.They are rational surfaces isomorphic to a projective plane blown up in 5 points with no 3 on a line, and are del Pezzo surfaces of degree 4, and have 16 rational...
s in P4, given by the intersection of two quadrics. They have 16 −1-curves. They form a 2-dimensional family.
Degree 5: they have 10 −1-curves corresponding to the minuscule vectors of one coset in the dual of the A4 lattice. There is up to isomorphism only one such surface, given by blowing up the projective plane in 4 points with no 3 on a line.
Degree 6: they have 6 −1-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 3 points not on a line. The root system is A2 × A1
Degree 7: they have 3 −1-curves. There is up to isomorphism only one such surface, given by blowing up the projective plane in 2 distinct points.
Degree 8: they have 2 isomorphism types. One is a Hirzebruch surface
Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .-Definition:The Hirzebruch surfaceΣn is the P1 bundle over P1associated to the sheafO+O.\...
given by the blow up of the projective plane at one point, which has 1 −1-curves. The other is the product of two projective lines, which is the only del Pezzo surface that cannot be obtained by starting with the projective plane and blowing up points. Its Picard group is the even 2-dimensional unimodular indefinite lattice II1,1, and it contains no −1-curves.
Degree 9: The only degree 9 del Pezzo surface is the P2. Its anticanonical embedding is the degree 3 Veronese embedding into P9 using the linear system of cubics.
Weak del Pezzo surfaces
A weak del Pezzo surface is a complete non-singular surface with anticanonical bundle that is nef and big.The blowdown of any −1-curve on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 more.
The blowup of any point on a weak del Pezzo surface is a weak del Pezzo surface of degree 1 less, provided that the point does not lie on a −2-curve and the degree is greater than 1.
Any curve on a weak del Pezzo surface has self intersection number at least −2. The number of curves with self intersection number −2 is at most 9−d, and the number of curves with self intersection number −1 is finite.
See also
- The mysterious duality relates geometry of del Pezzo surfaces and M-theoryM-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
. - Coble surfaceCoble surfaceIn algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system |−K| and non-empty anti-bicanonical linear system |−2K|. An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of...