This document presents notes from a seminar talk on Hecke curves and moduli spaces delivered by Heinrich Hartmann in 2008. It discusses Fano manifolds, particularly the moduli space SU(2,1) for vector bundles over curves of genus g ≥ 2, analyzing their construction and properties, including the existence of minimal rational curves known as Hecke curves. The document covers various mathematical concepts such as stability of vector bundles, Poincaré bundles, and the implications of their properties in the context of algebraic geometry.

1. Hecke Curves
Heinrich Hartmann
July 1, 2008
Abstract
This are the notes of a seminar talk delivered in summer 2008 at Bonn. Let SU(2, 1) be the
moduli space of rank 2 bundles with a fixed determinant of rank 1 over a curve C of genus g ≥ 2.
This is a Fano manifold of Picard rank 1. We discuss the example g = 2 where SU(2, 1) is the
intersection of two quadrics in P5
. In this case the minimal rational curves are lines. There is a
very interesting class of rational curves on SU(2, 1), called Hecke curves, which are constructed
by extending a given bundle by torsion sheaves. In the case g ≥ 3 we will see that Hecke curves
have minimal anti-canonical degree (4) and that any rational curve passing through a generic
point is a Hecke curve.
1 Moduli spaces of vector bundles
Let us recall some basic facts about moduli spaces. The main reference is LePotiers book [LEP].
1.1 Stability
Let F be an algebraic vector bundle of rank r > 0 and degree d on a smooth curve C over the
complex numbers C. The rational number
µ(F) :=
d
r
is called the slope. F is called stable (resp. semi-stable) if for all non-zero sub-bundles G ⊂ F the
inequality
µ(G) < µ(F) resp. µ(G) ≤ µ(F)
holds.
Remark 1. Line bundles are always stable. If rank and degree are coprime then stability and
semi-stability are equivalent.
1.2 Basic Properties
Let C be a smooth curve of genus g over C. The moduli functor of vector-bundles of rank r and
degree d is denoted by U(r, d). The value on a scheme S is the set of isomorphism classes of vector
bundles on S × C of rank r and degree d which are stable on each fiber {s} × C.
Theorem 2. U(r, d) is co-represented by an irreducible, projective variety U(r, d). Points in
U(r, d)(C) correspond to S-equivalence classes1 of semi-stable bundles. There exists an open subset
Us(r, d) ⊂ U(r, d) which parametrizes stable bundles.
1
See [HL] or [LEP] for the definition.
1

2. This means there is a morphism of functors ϕ : U(r, d) −→ hU(r,d) := Hom( , U(r, d)) which has
the following universal property: For each scheme Z and morphism of functors a : U(r, d) −→ hZ
there is a unique morphism of schemes b : U(r, d) −→ Z such that hb ◦ ϕ = a.
As always the pair (U(r, d), ϕ) is determined up to unique isomorphism by the universal property.
Remark 3. Tensoring with a line bundle induces isomorphisms between various moduli spaces.
Recall that for vector bundles F, G of rank r, s and degrees d, e respectively we have:
det(F ⊗ G) = det(F)⊗s
⊗ det(G)⊗r
It follows deg(F ⊗ G) = re + sd and deg(F ⊗ L) = d + r deg(L). Since we have plenty line bundles
of any degree ν ∈ Z (e.g. OC(ν · c), c ∈ C) we see:
U(r, d) ∼= U(r, d + rν)
for all ν ∈ Z. So we can always assume 0 ≤ d ≤ r − 1. (There may be problems with the stability
of F ⊗ L ?!)
Theorem 4. Us(r, d) is smooth of dimension r2(g − 1) + 1 (maybe empty). The tangent space at
a point [F] ∈ Us(r, d) is canonical isomorphic to Ext1(F, F).
Theorem 5. If the genus g of C is at least 2, there are stable bundles of any rank > 0 and degree.
In the case r, d coprime the moduli space has especially nice properties. From the last theorems
we see that U(r, d) = Us(r, d) is a smooth, projective variety. Moreover we have the following
Theorem 6. If r and d are coprime then there is a Poincare bundle on U(r, d) × C. That is
a bundle P ∈ U(r, d)(U(r, d)) such that the induced morphism ϕ(P) : U(r, d) → U(r, d) is the
indentity.
The Poincare bundle is not unique since for any line-bundle L on U(r, d) the tensor product
P ⊗ pr∗
1(L) has the same property.
1.3 Fixed Determinant
In the special case r = 1 any bundle is stable, so the moduli space of all line bundles of fixed degree
d exists and is a smooth projective variety of dimension g called Picd(C) := U(1, d). The tensor
product induces a group structure on J(C) := Pic0(C) witch makes J(C) an abelian variety called
the Jacobian of C. Note that J(C) ∼= Picν
(C) by Remark 3.
Let F be a vector bundle of rank r and degree d the determinant det(F) = Λr(F) is a line-bundle
of degree d. Since det is natural in F we get an induced map on moduli spaces:
det : U(r, d) −→ Picd
(C).
For a line bundle L let SU(r, L) be the fiber of this morphism. The tangent map at a point
[F] ∈ U(r, d) is given by
Ext1
(F, F)
tr
−→ H1
(C, OC) = Ext1
(det(F), det(F)).
This map is surjective since (1/r-times) the diagonal embeding OC → Hom(F, F) induces a section
H1(OC) → H1(Hom(F, F)) = Ext1
(F, F). Hence
Proposition 7. The map det : U(r, d) −→ Picd
(C) is submersive. If r, d are coprime SU(r, L) is
smooth of dimension (r2 − 1)(g − 1). The tangent space at a point F is canonically isomorphic to
H1(C, ad(F)) the trace free part of Ext1
(F, F).
2

3. Remark 8. Let L, L be line bundles of the same degree d. Since J(C) is a divisible group we can
always find an r-th root K of their difference: L ⊗ L
∨
= K⊗r. Now K induces an isomorphism
SU(r, L) ∼= SU(r, L ).
so the definition SU(r, d) := SU(r, L) makes sense.
Theorem 9 ([NR2]). If the genus g of C is at least 2 then SU(2, 1) is Fano of Picard number one.
Example 10. In the case g = 2 the moduli space SU(2, 1) is the intersection of two smooth
quadrics in P5, see [NR2].
In this case we can easily determine the variety of minimal rational tangents. Let X = Q1∩Q2 ⊂
P5 be the intersection of two quadrics. We claim there are 4 lines passing through a generic point
p ∈ X. This can be seen as follows:
Any line l ⊂ Q1 meeting p has to lie in the (projective) tangent space TpQ1 ⊂ P5. Therefore
we shall investigate Q1 ∩ Q2 ∩ TpQ1 ∩ TpQ2.
For any point q ∈ Q1 ∩ TpQ1 the line pq meets Q1 in two points p and q. The multiplicity of
the intersection at the point p is clearly greater 1. Since l.Q1 = 2 we find pq ⊂ Q1 hence Q1 ∩TpQ1
is a cone with vertex at p.
The cone Q1 ∩ TpQ1 is smooth away from p: Let q ∈ Q1 ∩ TpQ1 be a singular point. This
means the tangent space TqQ1 meets TpQ1 not transversally. But both are hyperplanes, hence
they are equal. Recall that the polar form of Q1 the unique quadratic form q1 with the property
Q1(x) = q1(x, x) for all x. One can check that the tangent space to Q1 has the following handy
description: TxQ1 = {[v] | q1(x, v) = 0} ⊂ P5. Since q1 is a non degenerate (⇔ Q1 smooth) it
follows immediately p = q.
If we intersect further with TpQ2 we get a cone over a smooth quadric in TpQ2 ∩ TpQ1
∼= P3.
Analogously we see Q2 ∩ TpQ2 ∩ TpQ1 is a cone over smooth quadric in the same projective space
with the same vertex. Therefore Q1 ∩ Q2 ∩ TpQ1 ∩ TpQ2 is the union of four lines.
2 Hecke Curves
2.1 Construction
Definition 11. Let F be a stable bundle of rank 2 and degree 1. So any proper sub bundle has
degree < 1/2 i.e. ≤ 0. We call F is strongly stable if there is no sub bundle of degree 0.
Proposition 12. If g ≥ 3, and L is a line bundle of degree 1 then a generic point [F] ∈ SU(2, L)
corresponds to a strongly stable bundle.
Proof. Let D be a degree 0 sub line bundle of a stable bundle F. The cokernel D = F/D is again
a line bundle. Since det(F) = L we have D ∼= L ⊗ D
∨
so F is the extension of a degree 0 line
bundle D with L ⊗ D
∨
.
Hence the bundles admitting degree 0 sub bundles are parametrized by the union of all
P(Ext1
(L ⊗ D
∨
, D)) = P(H1
(L∨
⊗ D2
))
as D varies in Pic0
(C). By Riemann Roch we see that this is a space of dimension h0(L∨ ⊗ D2) +
deg(L) + (g − 1) − 1 + g = 2g − 1 (h0 = 0 for a negative bundle) which is strictly smaller than
dim SU(2, L) = 3g − 3 whenever g ≥ 3.
Make rigorous by considering a Poincare bundle/Pic0 ×X and relative versions of Ext etc.
3

4. Definition 13. Choose a strongly stable rank 2 vector bundle F on C, a point c ∈ C and a line
l ∈ P(Fc). The elementary transformation ElF of F along l is defined by the exact sequence of
coherent sheaves on C:
0 → ElF → F → Fc/l ⊗ Oc → 0.
Note that ElF is a locally free sheaf of rank 2 and determinant det(F) ⊗ O(−c). We can apply the
same procedure to the dual of ElF to get a bundle with determinant det(F):
Set V := (ElF∨)c, for any choice of another line k ∈ PV ∼= P1 we define Ek
l F := (Ek(ElF∨))∨. So
we have
0 → Ek
l F∨
→ ElF∨
→ V/k ⊗ Oc → 0.
The map h : P1 → SU(2, 1), k → [Ek
l ] is called the Hecke curve associated to (F, c, l). That this is
well-defined and a morphism of schemes follows from the following lemmas.
Lemma 14. Let F be any rank 2 bundle set
m(F) := max{deg(L) | L ⊂ F line bundle } = max{deg(S) | S ⊂ F rank-one sub sheaf }.
This number is finite and m(F∨) = m(F) − deg(F) holds.
Proof. By Serre’s theorem we find k >> 0 s.th. F∨ ⊗ O(k · c) is globally generated for a c ∈ C, but
then there are no negative quotients. So m(F) ≤ k hence finite. Now let L ⊂ F∨ have maximal
degree and set K := F/L. We get K∨ ⊂ F hence − deg(K) ≤ m(F) and
m(F∨
) = deg(L) = − deg(K) − deg(F) ≤ m(F) − deg(F).
By symmetry we find m(F) ≤ m(F∨) + deg(F) and we get the required equality.
Lemma 15. If F is a strongly stable rank 2 degree 1 vector bundle then Ek
l F is stable for all
c ∈ C, l ∈ P(Fc), k ∈ P((ElF∨)c).
Proof of Proposition. We have to show m(Ek
l F) ≤ 0. Since ElF ⊂ F and F is strongly stable we
see m(ElF) ≤ m(F) ≤ −1. We apply the Lemma to the degree-0 bundle ElF to see m(ElF∨) =
m(ElF) ≤ −1. Now Ek
l F∨ ⊂ ElF∨ so m(Ek
l F∨) ≤ −1. Apply the lemma again to the degree-(−1)
bundle Ek
l F∨ to see m(Ek
l ) ≤ 0 as required.
Lemma 16. The map h : P1 → SU(2, 1), k → [Ek
l ] is a morphism of schemes.
Proof. By the universal property of SU(2, 1) we have to construct a vector bundle H on P1 × C
which restricts to [Ek
hF] on each fiber: Let c ∈ C, l ∈ Fc, V := (ElF∨)c. We have a universal
sequence on PV = P1:
0 → S → OP1 ⊗ V → OP1 (1) → 0.
Define H as the dual of the kernel of the epimorphism on P1 × C:
pr∗
2 ElF∨
→ pr∗
2(ElF∨
⊗ O{c}) ∼= V ⊗ OP1×{c} → pr∗
1 OP1 (1) ⊗ OP1×{c}
One checks easily that H has the desired properties.
Lemma 17. Every Hecke curve h : P1 → SU(2, 1) constructed from F passes through [F].
Proof. Restricting the defining sequence of ElF to the point c we get a exact sequence of vector
spaces:
0 → k → (ElF)c → Fc → (Fc)/l → 0
Now (ElF)c and Fc are both two-dimensional. Hence we get a one-dimensional kernel which we
denoted by k . The dual line defines a point k in PV . One checks that Ek
l F is isomorphic to F.
4

5. We won’t need this: More globally we can construct a morphism from a P1
-bundle H → PF to
SU(2, L) in such a way that the fiber over a point l ∈ PF parametrizes the corresponding Hecke curve.
Let us construct a relative version of the elementary transformation first. On PF there is a universal
sequence:
0 → SPF → π∗
(F) → OPF (1) → 0
We introduce another copy of C and arrive at the following situation:
C
π
←− PF
pr1
←−− PF × C
pr2
−−→ C
Now the γ := (idPF , π) : PF
∼=
−→ Γ ⊂ PF × C embeds PF as a divisor Γ in the product PF × C (the graph of
π). The restriction of the pullbacks pr∗
2 F and pr∗
1 π∗
F to this graph are isomorphic. So we get a canonical
surjection of sheaves on PF × C:
pr∗
2 F → pr∗
2 F ⊗ OΓ
∼= pr∗
1 π∗
F ⊗ OΓ → pr∗
1(QPF ) ⊗ OΓπ
We denote the kernel of this morphism by EF. By construction EF is a rank 2 vector bundle on PF × C
which restricts to the elementary transformation ElF over the fiber {l} × C of pr1.
We can now construct the parameter space H for our Hecke curves. We pull back EF∨
to PF via the
graph embeding γ : PF → PF × C and define H := P(γ∗
(EF∨
))
ρ
−→ PF. Again we get a universal sequence
on H:
0 → SH → ρ∗
γ∗
EF∨
→ OH(1) → 0
As before we see that on the product H × C the pullback pr∗
1 ρ∗
γ∗
EF∨
coincides with (ρ, idC)∗
EF∨
on the
graph ∆ of the projection ρ ◦ π : H → PF → C. Hence we get a surjection:
(ρ, idC)∗
EF∨
→ δ∗
EF∨
⊗ O∆
∼= pr∗
1 ρ∗
γ∗
EF∨
⊗ O∆ → pr∗
1(OH(1)) ⊗ O∆
The kernel of this morphism is restricts to Ek(Eρ(k)F∨
) on each fiber {k} × C. Hence the dual of this kernel
defines the required morphism:
H : H −→ SU(2, 1).
2.2 Minimality
Proposition 18. Hecke curves have anti-canonical degree 4.
Proof. The tangent space to SU(2, 1) at a point [F] given by a stable bundle F is naturally iso-
morphic to H1(C, ad(F)) ⊂ Ext1(F, F). On SU(2, 1) × C we have a Poincare bundle P. Note
that R0pr1∗ ad(P) = 0 since there are no traceless endomorphisms of a stable bundle. Hence
R1pr1∗ ad(P) is locally free with fibers isomorphic to T[F]SU(2, 1). In this situation general defor-
mation theory gives an isomorphism
TSU(2, 1)
∼= - R1
pr1∗ ad(P).
Let H be a vector bundle on P1 × C defining a Hecke Curve h : P1 → SU(2, 1). We saw
above that canonically h∗(TSU(2, 1)) ∼= h∗(R1pr1∗ ad(P)). By flat base change (in the version
of [MUM]) we can identify h∗(R1pr1∗ ad(P)) with R1 pr1∗(h × idC)∗ ad(U). Would the functor
defining SU(2, 1) have been representable we could conclude h∗(P) ∼= E, but SU(2, 1) is just a fine
moduli space. Nevertheless one may show [RAM] that in this situation h∗(P) ∼= H ⊗ pr∗
1 K for
some line bundle K on P1 and hence h∗(ad(P)) ∼= ad(H). So we get finally
h∗
(TSU(2, 1)) ∼= R1
pr1∗(ad(H)).
We calculate c1 using Grothendick-Riemann Roch:
c1(R1
pr1∗(ad(H))) = pr1∗(4c2(H) − c1(H)2
)
5

6. Now H is defined by a sequence of sheaves on P1 × C:
0 → H∨
→ pr∗
2 ElF∨
→ pr∗
1 O(1) ⊗ OP1×{c} → 0
We calculate ch(pr∗
2 ElF∨) = 1, ch(pr∗
1 O(1) ⊗ OP1×{c}) = (1 + t).(0 + s) = s + t.s, so ch(H∨) =
1 − s − ts where we denoted by s, t the generators of H2(P1, Z), H2(C, Z) respectively. Hence
c1(H)2 = 0 and c2(H) = ts, so we get
deg(h) := −KSU(2,1).[h] =
P1
c1(R1
pr1∗(ad(H))) =
P1×C
4c2(H) − c1(H)2
= 4
Proposition 19. Let [F] ∈ SU(2, 1) be generic then all rational curves through [F] have (anti-
canonical) degree at least 4.
Proof (following [SUN]). By proposition 12 we can assume F to be strongly stable. Let e : P1 →
SU(2, 1) be a rational curve defined by a bundle E on P1 × C. Recall from proof of Proposition 18
that
deg(e) = −KSU(2,1).[e] = 4c2(E) − c2
1(E).
Thus we shall compute ci(E): As
H2
(P1
× C) = H2
(P1
) ⊕ H2
(C) =: Z < t, s >
(by K¨unneth and H1(P1) = 0) we can calculate c1(E) from the degrees of the restrictions to
generic fibers of the two projections: Fix generic points x ∈ P1, c ∈ C and set Ec := E|P1×{c} resp.
Ex := E{x}×C then
c1(E) = deg(Ec)t + deg(Ex)s.
Now deg Ex = 1 since we are mapping to SU(2, 1), and deg(Ec) can be assumed to be anything
since tensoring by pr∗
1 OP1 (k) does not effect h. Let us look a bit closer: Ec is a line bundle on P1
and hence splits as O(a) ⊕ O(b), w.l.g. a ≥ b and b = 0 by the same argument.
Case a = 0 i.e. Ec
∼= O2. So c2
1(E) = 0 and we should show c2(E) ≥ 1. This follows from the
following:
Lemma 20. Let E be a torsion free sheaf of rank r on P1 × C with Ec
∼= O⊕r
P1 for a generic c ∈ C,
then c2(E) ≥ 0 and c2(E) = 0 iff E = pr∗
2 F for a locally free sheaf F on C.
Proof of Lemma. By Induction to r. Suppose r = 1. Consider the sequence of sheaves:
0 → E → E∨∨
→ T → 0.
It is E∨∨ reflexive and we are on a surface, so it is locally free, moreover T is supported at a finite
set of points. Now c2(E) = h0(T) ≥ 0 by Riemann Roch so if c2(E) = 0 then T = 0 so E ∼= E∨∨
is a line bundle. The Picard group of P1 × C is isomorphic to Pic(P1) × Pic(C) and since Ec
∼= O
on a generic fiber, E has to be the pull-back of a bundle F on C.
r > 1 : We find a sub sheaf L ⊂ E such that E/L is torsion free and splits as O⊕n−1 along a generic
fiber of pr2. Indeed take any saturated sub line bundle L of pr2∗ E, the image of the pullback
pr∗
2 L ⊂ pr∗
2 pr2∗ → E to is a line bundle L with E/L = pr2∗ By induction c2(E/L) ≥ 0 and = 0 iff
E/L is the pull back of a locally free sheaves on C. We have
c2(E) = c2(L) + c2(E/L) + c1(L).c1(E/L) = c2(L) + c2(E/L) ≥ 0
So c2(E) = 0 iff c2(E/L) = c2(L) = 0 and in this case both are pullbacks of locally free sheaves by
induction hypothesis. We conclude that Ec
∼= Or for all c ∈ C. And hence E is a pullback of a
6

7. sheaf F on C. ( Indeed h0(P1, Ec) = h0(P1, O2) = 2 is a constant function of c, so pr2∗E is locally
free and the fiber at c is H0(P1, Ec) by Grauerts theorem. We have a canonical map of vector
bundles
pr∗
2 pr2∗ E → E
by adjunction. Using flat base change its easy to see, that this is an Isomorphism. )
Proof of Proposition. Continuation. We are left with the Case a > 0:
The relative Hader Narasimhan filtration2 of pr2 gives a tool to lift the decomposition Ec = O(a)⊕O
on the generic fiber to the product P1 × C. It is a filtration
0 = E0 ⊂ E1 ⊂ E2 = E
with the properties: F1 = E1, F2 = E/E1 are torsion free and and restrict to O(a), O respectively
on a generic fiber. We calculate
deg(e) = 4c2(E) − c2
1(E) = 4(c2(F1) + c2(F2)) + 2c1F1.c1F2 − c1F2
1 − c1F2
2 .
We get the first Chern classes of Fi by taking the degree on the generic fibers:
c1(F1) = d1t + as, c1(F2) = d2t
where d1 +d2 = 1. Note that Ec is a strongly stable bundle for generic c ∈ C and since F1 = E1 ⊂ E
we know d1 ≤ −1. Substituting above yields:
deg(e) = 4(c2(F1) + c2(F2)) + 2(ad2)t.s − 2(ad1)t.s − 0
= 4(c2(F1) + c2(F2)) + 2a(1 + 2(−d1))t.s
≥ 4(c2(F1) + c2(F2)) + 6ts.
Now one checks that c2(F1) = c2(F1 ⊗ pr∗
1(−a)). So we can apply the above lemma to finish the
proof.
Proposition 21. If e : P1 → SU(2, 1) is a rational curve of minimal degree (= 4) passing through
a generic point [F] then e is a Hecke curve.
Proof. We use the notation as above. In the proof of proposition 19 we saw that a curve has
minimal degree iff Ec splits as O⊕2
P1 for generic c ∈ C and c2(E) = 1.
There is at least one c0 ∈ C such that, Ec0
∼= O(a)⊕O(b) with a > b, otherwise we see as in lemma
20 that E is the pullback of a bundle on C, so e is constant - a contradiction. Since the degree of
Ec is zero for generic c ∈ C we have a + b = 0. Consider the surjection:
E → E ⊗ OP1×{c0} → OP1×{c0}(b)
The kernel K of this map is a torsion free sheaf of rank 2 and satisfies c2(K) = c2(E) + b. As
c2(K) ≥ 0 by the previous lemma we must have b = −1 and c2(K) = 0 and so K = pr∗
2(F) for
some vector bundle F on C (again by lemma 20). So we arrived at the following situation:
0 → pr∗
2(F) → E → OP1×{c0}(−1) → 0
dualizing yields:
0 → E∨
→ pr∗
2(F)∨
→ OP1×{c0}(1) → 0
(because OP1 (1) = Ext1
(OP1 (−1), OP1 )) but this means precisely e is a Hecke curve.
2
See [HL] for details.
7

8. Proposition 22. Hecke curves are smooth. Different choices of h ∈ PF define different curves.
Proposition 23. Hecke curves are free rational curves.
See [NR] or [CAS] for proofs.
Proposition 24. Let [F] ∈ SU(2, 1) be generic, the minimal rational component K[F] ⊂ Hilb(C)
consists of Hecke curves and hence is naturally isomorphic to P(F).
This is an easy corollaray form what was said above, see [HWT]. There is an explicit description
of τ : P(F) → P(T[F]SU(2, 1)) in [HW-R].
References
[HL] D. Huybrechts, M. Lehn, The Geometry of Moduli Spaces of Schemes, 1996
[LEP] J. LePotier, Lectures on vector bundles, Cambridge University Press, 1997
[MUM] D. Mumford, Lectures on Curves on an algebraic surface, Princeton University Press, 1966
[RAM] S. Ramanan, The Moduli Space of Vector bundles over an Algebraic Curve, Math. Ann.
200, 1973
[NR] M. S. Narasimhan, S. Ramanan, Deformations of the moduli space of vector bundles over
an algebraic curve, 1973
[NR2] M. S. Narasimhan, S. Ramanan, Moduli of vector bundles on a compact Riemann surface,
Ann. Math. 89, 1969
[CAS] A.-M. Castravet, Rational families of vector bundles on curves, I, arXiv:math/0302133,
2003
[HWT] J.-M. Hwang, Tangent vectors to Hecke curves on the Modulispace of rank 2 bundles over
an algebraic curve, Duke Math. Journal Vol. 101, No. 1, 2000.
[HW-R] J.-M. Hwang, S. Ramanan, Hecke curves and Hitchin discriminant, arXiv:math/0309056,
2003
[SUN] X. Sun, Minimal rational curves on moduli spaces of stable bundles, Math. Ann. 331, 2005
8