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Complex Algebraic Surfaces, Homework Assignment 3, Spring 2009
1 Introduction
Let R := C[x] be the ring of polynomials. Let Vn,d be the vector space of all n×n matrices with entries in R, such that the degree of each entry is ≤ d. Clearly, dim(Vn,d) = n^2 (d+1). Given a matrix A = (aij (x)) in Vn,d, its characteristic polynomial
charA(x, λ) := det[A − λI]
is a polynomial in two variables. The zero locus of charA(x, λ) is an affine plane curve, called the affine spectral curve of A. Algebraic curves very often arise in other branches of mathematics as spectral curves (see [B2] for examples arising in classical mechanics). In problem 3 below you will prove the following statement, for all d ≥ 1 and n ≥ 1. Set Fd := P(OP 1 (d) ⊕ OP 1 ) and let p : Fd → P^1 be the natural morphism. Set M := OPE (1) ⊗ p∗OP 1 (d). Let Mn^ be the n-th tensor power of M.
Theorem 1 There exists a Zariski dense open subset of Vn,d, consisting of matrices A, whose affine spectral curve is a Zariski open subset of a smooth connected projective
curve C˜ of genus d
n(n−1) 2
− n + 1. The curve C˜ is naturally embedded^1 in the ruled
surface Fd as a divisor in the linear system |Mn|.
The construction introduces a morphism char : Vn,d → |Mn|. In Problem 4 you will
describe the fiber char−^1 ( C˜) in terms of the spectral curve C˜.
Set F := ⊕ni=1OP 1. Key to the proof is the observation that an element A of Vn,d corresponds to a homomorphism of OP 1 -modules ϕ : F → F ⊗ OP 1 (d) as follows. Choose homogeneous coordinates (t 0 , t 1 ) over P^1. Set ϕij (t 0 , t 1 ) := td 0 aij (t 1 /t 0 ). Then ϕij is a homogeneous polynomial of degree d, hence a section of H^0 (P^1 , OP 1 (d)). We get the isomorphism
Vn,d ∼= Hom(F, F ⊗ OP 1 (d)), (aij ) 7 → (ϕij ).
N. Hitchin discovered in the 1980’s that spectral curves play an important role in the study of n-dimensional irreducible complex representations of the fundamental group of a complex projective curve C of positive genus [H]. Hitchin’s pairs (F, ϕ) consist of a rank n vector bundle F on C and its “endomorphism” ϕ : F → F ⊗ ωC is twisted by the canonical line-bundle ωC. Hitchin’s spectral curves are embedded in the ruled surface P[ωC ⊗ OC ]. The genus of Hitchin’s spectral curve, which you will calculate below, is equal to half the dimension of the space of representations of the fundamental group. Terminology: A rank n vector bundle over an algebraic variety X is a locally free OX -module of rank n. The following three objects are one and the same: a line-bundle, an invertible sheaf, and a locally free OX -module of rank 1.
(^1) Note that the closure of such a curve in P (^2) has degree nd, so arithmetic genus (nd − 1)(nd − 2)/2.
The latter is larger than the geometric genus by n(d − 1)[nd − 2]/2. Hence the closure in P^2 is singular, except possibly when d = 1, or (n, d) = (1, 2).
2 Problems
- Let C be a smooth curve, L a line bundle on C of degree d, E := L ⊕ OC , and p : PE → C the corresponding ruled surface. The line sub-bundle L of E corresponds to a section σ∞ : C → PE, whose image is Σ∞ := PL. Let σ 0 : C → PE be the section corresponding to the line sub-bundle OC of E, and denote its image by Σ 0. The fiber of [PE \ Σ∞] over y ∈ C can be naturally identified with the fiber Ly of L, and σ 0 (y) is its zero point. Simply associate to ℓ ∈ Ly the point in PE corresponding to the line spanC{(ℓ, 1)} in the fiber of E.
(a) Show that Σ 0 belongs to the linear system |(p∗L) ⊗ OPE (1)| and Σ∞ belongs to |OPE (1)|. Hint: Consider the tautological exact sequence
0 → OPE (−1) → p∗(E) → QPE → 0.
Show that the section (0, 1) of p∗E maps to a non-zero section of QPE , which vanishes along Σ 0 with multiplicity 1. Then repeat your argument for the section (1, 0) of p∗(E ⊗ L−^1 ). (b) Let D ⊂ PE be an irreducible curve, which is disjoint from Σ∞. Show that the class [D] of D in H^2 (PE, Z) is n(df + h), where f is the class of the fiber, h := c 1 (OPE (1)), and n := ([D], f ). Conclude that the arithmetic genus of D
is g(D) = d
n(n − 1) 2
- n[g(C) − 1] + 1. Caution: In Proposition III.18 in Beauville’s text [B1] his OS (1) is our QPE.
- Keep the notation of problem 1. Set M := (p∗L)⊗OPE (1). Following is an explicit construction of smooth curves in the linear system |Mn|, which are disjoint from Σ∞. Choose bi ∈ H^0 (C, Li), 0 ≤ i ≤ n. Set b := (b 0 , b 1 ,... , bn) and ai := p∗bi. Choose a section λ 1 of H^0 (PE, OPE (1)), with divisor Σ∞ (λ 1 is unique, up to a scalar factor). If we identify OPE (1) with OPE (Σ∞), then λ 1 can be the section 1 of the latter. Choose a section λ 0 of H^0 (PE, M), with divisor Σ 0. We get the section σb :=
∑^ n
i=
aiλi 1 λn 0 −i ∈ H^0 (PE, Mn). (1)
Denote by C˜b the divisor in |Mn| corresponding to σb.
(a) Show that if b 0 6 = 0, then the intersection C˜b ∩ Σ∞ is empty. (b) Show that if b 0 6 = 0, bi = 0, for 1 ≤ i ≤ n − 1, and the divisor of bn in |Ln| consists of nd distinct points of C, then the curve C˜b is smooth and irreducible. Note: Points in a linear system, corresponding to smooth divisors, form a Zariski open subset (see Hartshorne’s Algebraic Geometry, Ch. I, section 5, Problem 5.15). (c) Prove that H^0 (PE, Mn) decomposes as the direct sum ⊕ni=0λi 1 λn 0 −ip∗H^0 (PE, Li). Conclude that every section of H^0 (PE, Mn) is of the form given in Equa- tion (1). Hint: It suffices to establish the direct sum decomposition
H^0 (PE, Mk) = λ 0 H^0 (PE, Mk−^1 ) ⊕ λk 1 p∗H^0 (C, Lk),
is an isomorphism. Hint: It suffices to prove injectivity, by part 4c. See Remark 2 for the meaning of this isomorphism.
Remark 2 When C˜ is smooth, the sheaf F˜ is a locally free O (^) Ce -module of rank 1, by part 4d. The isomorphism class of F˜ determines the isomorphism class of the pair (F, ϕ), and so the P GL(n, C)-orbit of the matrix A ∈ Vn,d, as follows. Let μ : F˜ → F˜ ⊗ M be the homomorphism, given by tensoring with the section λ 0 of M. The push-forward p∗(μ) is equal^3 to the homomorphism ϕ : F → F ⊗ L, up to conjugation of ϕ by an automorphism of F. Set d˜ := χ( F˜ ) + 1 − ˜g. The algebraic variety Pic d˜ ( C˜), of degree d˜ line-bundles on C˜, is a ˜g-dimenstional smooth algebraic variety (Its dimension is equal to h^1 (C, OC ), by the discussion in Section I.10 of Beauville’s text on the exponential sequence [B1]). Hence, the fiber char−^1 ( C˜) is an algebraic subset of Vn,d of dimension at most ˜g + dim P GL(n, C). This must be exacltly the dimension of the fiber, by part 4a. See [BNR] for a detailed exposition.
- Do problems 1, 2, 5, 6 in Chapter III page 37 of Beauville’s text [B1].
References
[B1] Beauville, A.: Complex Algebraic Surfaces. Second Edition. London Math. Soc. Student Texts 34, Cambridge Univ. Press 1996.
[B2] Beauville, A.: Jacobiennes des courbes spectrales et systemes hamiltoniens completement int´egrables. Acta Math. 164, 211-235 (1990)
[BNR] Beauville, A., Narasimhan, M. S., Ramanan, S.: Spectral curves and the generalized theta divisor. J. Reine Angew. Math. 398, 169-179 (1989)
[H] Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55 (1987) 59–126.
(^3) The above statement is due to the fact that a fiber of F˜ over a point x of C˜ is naturally identified
with the x-eigen-line of the fiber F (^) p(x) of F over p(x), provided the eigenvalue x has multiplicity one (i.e., provided x is not a ramification point of C˜ → C). Furthermore, μ acts on this fiber of F˜ via tensorization with the corresponding eigenvalue x ∈ Lp(x). Finally, the fiber of p∗ F˜ over y ∈ C is naturally identified with the direct sum of the fibers of F˜ , over points in p−^1 (y), provided y is not a branch points of C˜ → C.
