Geometric realization of root systems and theJacobians of del Pezzo surfaces
Shigeru Mukai ∗In the conference talk, I reviewed the geometric part of my article [2] andannounced the following result on the Jacobian of a del Pezzo surface, whosedetails will be published elsewhere.
A smooth complete algebraic surface S (over an algebraically closed field)
is del Pezzo if the anti-canonical class −KS is ample. The self-intersection
number (−KS2 ) =: d is called the degree, which ranges from 1 to 9. A delPezzo surface S is isomorphic to the projective plane P2 if d = 9, and toeither a smooth quadric surface Q or the blow-up of P2 at a point if d = 8.
A del Pezzo surface S of degree d ≤ 7 is isomorphic to the blow-up of P2 at(9 − d) points in a general position.
The anti-canonical system |−KS | is of dimension d and its general member
is a smooth elliptic curve. Let C ⊂ S × Pd be the universal family of anticanonical members C ∈ | − KS | = Pd and Cη be the generic fiber of C −→ Pd .
Definition A morphism ϕ : J˜ −→ Pd is a Jacobian fibration of S if all fibers
are of dimension one and its generic fiber is the Jacobian of the generic anticanonical member Cη . A (d + 1)-dimensional variety J with a smooth point
p is a reduced Jacobian of S if the blow-up of J at p have a Jacobian fibration
ϕ such that the exceptional divisor over p is the 0-section of ϕ.
In the case of degree d = 1, the anti-canonical system | − KS | is a pencil
with a unique base point. Hence a del Pezzo surface S itself is its reducedJacobian.
Theorem For a del Pezzo surface S, there exists a reduced Jacobian (J(S), p)
which satisfies the following properties:(1) J(S) is a (d + 1)-dimensional weak del Pezzo variety of degree one,that is, −KS = dH, H being a nef and big divisor with (H d+1 ) = 1,∗

Supported in part by the JSPS Grant-in-Aid for Scientific Research (B) 17340006.

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(2) p is the unique base point of the d-dimensional linear system |H|,
(3) J(S) is the blow-up of the projective space P3 at seven points in ageneral position if d = 2,(4) J(S) is the blow-up of the product P2 × P2 at five points in a generalposition if d = 3,(5) J(S) is the blow-up of the 6-dimensional Grassmannian G(2, 5) atfour points p1 , . . . , p4 in a general position if d = 5, and(6) J(S) is the blow-up of a singular hyperplane section G(2, 5)′ of G(2, 5) ⊂P9 at four points p1 , . . . , p4 in a general position if d = 4.
In the case d = 1, 2, 3, the reduced Jacobian J(S) belongs to the class ofrational varieties studied in [2]. The augmented root system N (E9−d , adjoint)(cf. [3, §4] and [4, §4]) is realized in the second cohomology group H 2 (J(S), Z).
The Weyl group W (E9−d ) birationally acts on the universal family of J(S)over the configuration space of (9 − d) points on P2 . Similar properties holdtrue for a del Pezzo surface Sd of degree d = 5, 4 with the following augmented
root system.
e1 − e2e2 − e3

h − 2e1 − e2 − e3 − e4d = 5,e3 − e4

e1 − e2e4(A4 , adjoint)e2 − e3e3 − e4h 1 − e1 − e2e4

d = 4,

h 2 − e1 − e2 − e3

(D5 , adjoint)

In these diagrams in H 2 (J(Sd ), Z), h denotes the pull back of a hyperplane section of the Plücker embedding G(2, 5) ⊂ P9 , and hi denotes that ofa divisor class of G(2, 5)′ with dim |hi | = i. e1 , . . . , e4 are the classes of exceptional divisors over p1 , . . . , p4 . The reflection with respect to the (−2)-classh − 2e1 − e2 − e3 − e4 ∈ H 2 (J(S5 ), Z) is realized by the composite of two birational involutions of J(S5 ) = Blp1 ,...,p4 G(2, 5). One is the Geiser involution
the blow-up of G(2, 5) at p2 , p3 and p4 , that is, the covering involution of themorphism
Φ|H−e2 −e3 −e4 | : Blp2 ,p3 ,p4 G(2, 5) −→ P6of degree 2. The other is the Bertini involution, that is, the involution of˜ 5 ) −→ P5 . (SeeJ(S5 ) induced from the (−1)J˜ of the elliptic fibration ϕ : J(S[1, §7] for the Geiser and Bertini involutions of del Pezzo surfaces.)

135References[1] Dolgachev, I.: Weyl groups and Cremona transformations, Proc. AMS
Symp. Pure Math. 40(1978), 283–294.
[2] Mukai, S,: Geometric realization of T-shaped root systems and counterexamples to Hilbert’s fourteenth problem, ‘Algebraic Transformation
Groups and Algebraic Varieties’, ed. V. L. Popov, Springer-Verlag, 2004,
pp. 123–129. (RIMS preprint 1372)[3] Kanev, V.: Spectral curves, simple Lie algebras, and Prym-Tjurin varieties, Proc. AMS Symp. Pure Math. 49(1989), Part I, 627–645.
[4] — : Spectral curves and Prym-Tjurin varieties I, ‘Abelian Varieties’, eds.
W. Barth, K. Hulek and H. Lange, Walter de Gruyter, Berlin · New York,
1995, pp. 151–198.

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