CHAPTER 4. MODULI OF PARABOLIC BUNDLES
4.2 The moduli stack of parabolic bundles over P 1
Definitions and general properties of algebraic stacks are given in Laumon and Moret-Bailly’s book [LMB]. We will view a stack as a sheaf of groupoids in the fppf-topology and an algebraic stack as a stack with a smooth presentation by a scheme. We will useh i to denote a category in which the objects are enclosed by the brackets and the morphisms are all isomorphisms.
As before, let X be the smooth complex projective curve. Fix the weight type (D, w) as in Section 4.1. Let I ={0} ∪ {(i, j)|1≤i≤k,1≤j≤wi−1}, let d∈Z , and fixα∈ZI≥0, such that α0 ≥αi1 ≥ · · · ≥αiwi, for all i.
Definition 4.2.1. The stack of parabolic bundles of weight type(D, w), degree d, dimension type α, overX is a functor that associates to a test scheme T the groupoid BundD,w,α(T) =(E, Ei,j)1≤i≤k
, where • E is a vector bundle on T×X, • E|T×{x i}⊃E
i,1⊃ · · · ⊃Ei,wi−1 ⊃Ei,wi = 0 is a filtration by vector bundles, • rk(E) =α0 and rk(Ei,j) =αij,
• degE|{y}×P1 =d for ally∈T.
In the case whenX =P1, we see that BundD,w,α admits the following presentation as an algebraic
stack: U =`
N∈Z≥0h(E, si, tj)i, where
• E is a parabolic bundle on X,
• deg(E) =dand E has dimension vector α, • H0(E∗⊗ O(N)) is generated by global sections, • si is a basis forH0(E∗⊗ O(N))∗,
• tj is a basis forH0(E∗⊗ O(N−1))∗.
For X=P1, we will give a more detailed description of U in Section 6. Let B:= BunD,w,α(X) =
`
d∈ZBundD,w,α be the moduli stack of parabolic bundles of weight type (D, w) and with dimension
vectorα. We can use the presentation above to turn this stack into an algebraic stack. Definition 4.2.2. The stack of pairs PBund
D,w,α is a functor that associates to a test scheme T the
groupoid PBund D,w,α(T) = (E, Ei,j, f)1≤i≤k , where • E is a vector bundle on T×X,
• E|T×{xi}⊃Ei,1⊃ · · · ⊃Ei,wi−1 ⊃Ei,wi = 0 is a filtration by vector bundles, • rk(E) =α0 and rk(Ei,j) =αij,
• degE|{y}×P1 =d for ally∈T,
• f is an endomorphism of E such that f|T×{xi}(Ei,j)⊂Ei,j for all i.
Before we give a presentation for PBund
D,w,α as an algebraic stack, we will need some preliminary notation. LetE, sl, tmbe as in the description ofU above. LetG0(sl, tm) andG1(sl, tm) be matrices
in the bases sl andtm, representing the morphisms from H0(E∗⊗ O(N))∗ toH0(E∗⊗ O(N −1))∗
that correspond to multiplication by 1, z, the two standard generating global sections ofO(1). Note that ker (λi0G0(sl, tm) +λi1G1(sl, tm)) contains the flag Exi = Ei0 ⊇ Ei1 ⊇ Eiwi−1 ⊇ Eiwi = 0, where xi = (λi0 :λi1) (see Section 6 for details).
In the case whenX=P1, we see thatPBund
D,w,α admits the following presentation as an algebraic stack: U =`
N∈Z≥0h(E, sl, tm, F1, F2)i, where
• E is a parabolic bundle on X,
• deg(E) =dand E has dimension vector α, • H0(E∗⊗ O(N)) is generated by global sections, • sl is a basis forH0(E∗⊗ O(N))∗,
• tm is a basis for H0(E∗⊗ O(N −1))∗,
• F1 is a matrix in the basis sl acting onH0(E∗⊗ O(N))∗, • F2 is a matrix in the basis tm acting on H0(E∗⊗ O(N −1))∗,
• F2◦Gr(sl, tm) =Gr(sl, tm)◦F1 forr = 0,1, • F1(Eij)⊂Eij for all 1≤i≤kand 1≤j≤wi.
Since our computations are independent of degree, we will define the algebraic stack PB := PBunD,w,α(X)=`
d∈ZPBundD,w,α.
Let IB = IBunD,w,α(X) be the inertia stack corresponding to BunD,w,α(X). This is an open substack ofPB, where the endomorphismf is taken to be an automorphism. Note that dimIB = dim PB.
The following stack will play an essential role in the proof of Theorem 1.2.1:
Definition 4.2.3. The stack N(D, w, d, α) is a functor that associates to a test scheme T the groupoid N(D, w, α)(T) =(E, Ei,j, f)1≤i≤k
, where
• E is a vector bundle on T×X,
• E|T×{xi}⊃Ei,1⊃ · · · ⊃Ei,wi−1 ⊃Ei,wi = 0 is a filtration by vector bundles, • rk(E) =α0 and rk(Ei,j) =αij,
• degE|{y}×P1 =d for ally∈T,
We can see that N(D, w, d, α) is a reduced closed algebraic substack of PB, given the following presentation (assuming thatX =P1): U =`N∈Z≥0h(E, sl, tm, F1, F2)i, where
• E is a parabolic bundle on X,
• deg(E) =dand E has dimension vector α, • H0(E∗⊗ O(N)) is generated by global sections, • sl is a basis forH0(E∗⊗ O(N))∗,
• tm is a basis for H0(E∗⊗ O(N −1))∗,
• F1 is a nilpotent matrix in the basis sl acting onH0(E∗⊗ O(N))∗,
• F2 is a nilpotent matrix in the basis tm acting on H0(E∗⊗ O(N −1))∗,
• F2◦Gr(sl, tm) =Gr(sl, tm)◦F1 forr = 0,1, • F1(Eij)⊂Eij for all 1≤i≤kand 1≤j≤wi.
Our computations are independent of degree, we will define the algebraic stack N(D, w, α) := `
d∈ZN(D, w, d, α).
Note that BunD,w,α(X) is smooth, and by Lemma 2.2.1 we can compute its dimension as:
dim BunD,w,α(X) = dim BunGL(α0)(X) + dimF l(α)
= (g−1)α20+α20−q(α) =gα20−q(α).
From now on, letX=P1. This meansg= 0, and therefore dim BunD,w,α(X) =−q(α).