Cox rings of K3 surfaces
Michela Artebani
Departamento de Matemática
Universidad de Concepción
elENA IV (La Falda, Sierras de Córdoba)
August 4, 2008
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Cox rings
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Outline
1
Cox rings
Definition
Relations
2
K3 surfaces
Motivations
K3 surfaces with finitely generated NE(X)
3
Cox rings of K3 surfaces
rk Pic(X) = 2
Non symplectic involutions
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Cox rings
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Cox rings
Let X be a smooth algebraic variety over C and L1 , . . . , Lρ be divisors
whose classes freely generate Pic(X).
The Cox ring of X is:
Cox(X) :=
M
H 0 (X, a1 L1 + · · · + aρ Lρ )
a∈Zn
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Cox rings
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Cox rings
Observation. Cox(X) is graded by Pic(X):
M
Cox(X) ∼
Cox(X)D
=
D∈Pic(X)
Cox(X)D = H 0 (X, D),
Cox(X)D1 · Cox(X)D2 ⊆ Cox(X)D1 +D2 .
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Cox rings
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Cox rings
Questions.
When is Cox(X) finitely generated?
How do we find its generators and relations?
0
/ IX
Michela Artebani (UdeC)
/ C[x1 , . . . , xr ]
Cox rings
/ Cox(X)
/ 0.
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Cox rings
Theorem (D. Cox, 1992). Let X be a toric variety with E1 , . . . , Er
integral invariant divisors, then
Cox(X) = C[x1 , . . . , xr ]
where xi is a defining section for Ei .
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Cox rings
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Cox rings
Example: Let F0 be a quadric surface:
a2
b2
b1
Cox(X) ∼
= C[a1 , a2 , b1 , b2 ]
a1
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Cox rings
Examples of smooth surfaces with finitely generated Cox ring:
toric,
Del Pezzo,
blow-up of P2 at points lying on a conic,
some K3 surfaces.
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Cox rings: relations
Theorem (A. Laface, M. Velasco, 2007). Let E1 , E2 be two generators
of Cox(X) with E1 · E2 = 0, if the map:
2
M
H 0 (X, D − Ek − Ei )
H 0 (X, D − Ei )
k=1
(a, b)
ax1 + bx2
is surjective for any i > 2 then IX has no generators in degree D.
Observation. This happens if
H 1 (X, D − E1 − E2 − Ei ) = 0
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Cox rings
for each i > 2.
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K3 surfaces
A K3 surface X is a
simply connected compact complex surface
which admits
a nowhere vanishing holomorphic two form ωX .
Examples:
Quartic surfaces of P3 .
Double covers of Del Pezzo surfaces branched along B ∈ | − 2KY |.
Remark: P ic(X) is a lattice.
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Cox rings
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K3 surfaces
Given an integral divisor D ⊂ X on a K3 surface, the graded algebra:
M
R(X, D) :=
H 0 (X, iD)
i∈Z
is finitely generated by elements of degree i ≤ 3.
Remark:
Cox(X) f.g implies that R(X, D) is f.g.
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K3 surfaces
Which K3’s have a f.g. Cox ring?
NE(X) must be finitely generated.
A theorem of Kovács (1994) describes the generators of NE(X):
either an ample class or rational curves with self-intersections 0
and −2.
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Cox rings
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Cox rings of K3 surfaces (with J. Hausen and A. Laface)
We deal with two classes of K3 surfaces:
with rk Pic(X) = 2,
admitting a non-symplectic involution.
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K3 surfaces with ρ = 2
X is a K3 surface with rk Pic(X) = 2 and cone of curves
NE(X) = hE1 , E2 i.
The intersection matrix of Pic(X) is:
0 k
k 0
0
k
k −2
= U (k) with k ≥ 2
−2
k
k −2
with k ≥ 1
with k ≥ 3
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K3 surfaces with ρ = 2
Theorem. Let X be a K3 surface with Pic(X) ∼
= U (k) and k ≥ 2, then
the degrees of generators and relations of Cox(X) are:
k
2
≥3
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deg. gen.
(1, 0), (0, 1), (2, 2)
(1, 0), (0, 1), (1, 1)
Cox rings
deg. rel.
(4, 4)
(2, 2)
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K3 surfaces with ρ = 2
Example. If X is a K3 surface with Pic(X) ∼
= U (2) then
Cox(X) ∼
=
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C[a1 , a2 , b1 , b2 , c]
.
(c2 − f4,4 (a, b))
Cox rings
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K3 surfaces with non-symplectic involution
Proposition. Let (X, σ) be a generic pair of a K3 surface together with
a non-symplectic involution and
π : X −→ X/ < σ >= Y
be the induced double cover. Then
Y is either smooth rational or an Enriques surface,
π ∗ Pic(Y ) has index 2r in Pic(X), where r − 1 is the number of
components of the branch locus of π.
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K3 surfaces with non-symplectic involution
Let R be the fixed locus of σ and B = π(R).
Lemma. For any divisor D of Y :
H 0 (X, π ∗ D) ∼
= π ∗ H 0 (X, D) ⊕ xR · π ∗ H 0 (X, D − B/2)
where xR is the section defining R.
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K3 surfaces with non-symplectic involution
Theorem. Let (X, Y, σ) as before. If the branch divisor B of π is
irreducible then
Cox(X) ∼
=
Cox(Y )[xR ]
.
(x2R − xB )
Corollary. Cox(X) is known if Y is a Del Pezzo surface.
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K3 surfaces with non-symplectic involution
ρ(X)
2
3
4
5
6
Michela Artebani (UdeC)
Pic(X)
U
U (2)
(2) ⊕ A1
U ⊕ A1
U (2) ⊕ A1
U ⊕ 2A1
U (2) ⊕ 2A1
U ⊕ 3A1
U (2) ⊕ 3A1
U ⊕ 4A1
U (2) ⊕ 4A1
U ⊕ D4
U (2) ⊕ D4
Y
F4
F0
P2
F4
F0
F4
F0
F4
F0
F4
F0
F4
F0
Cox rings
Blow-up at:
1 pt.
1 pt.
1 pt.
2 pts.
2 pts.
3 pts.
3 pts.
4 pts.
4 pts.
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References
V. Batyrev, O. Popov
The Cox ring of a del Pezzo surface.
Progr. Math. 226, Birkhauser Boston, Boston, MA, 85–103, (2004).
F. Berchtold, J. Hausen
Homogeneous coordinates for algebraic varieties.
J. Algebra 266(2), 636–670, (2003).
D. Cox
The homogeneous coordinate ring of a toric variety.
J. Algebraic Geom. 4(1), 17–50, (1995).
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References
S. Kovács
The cone of curves of a K3 surface.
Math. Ann., 300, 681 - 691, (1994).
A. Laface, M. Velasco
Picard-graded Betti numbers and the defining ideal of Cox rings.
arXiv:0707.3251v1
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