Some Properties of the Filtration \(\mathbf{\tilde{H}}_{-\bullet }\)
We fix a stratum \(\Gamma _{d}^{r}\) in (2.13) according to the conventions in §2.7 and consider an admissible component \(\Gamma \) in \(C_{adm}^{r}(L,d)\) .
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Q Some Properties of the Filtration H
We fix a stratum dr in (2.13) according to the conventions in 2.7 and consider an r admissible component in Cad m .L; d /. Lemma 3.1. Let ŒZ 2 and let JZ D 1 .ŒZ/ be the fibre of over ŒZ. Denote .0/ by JZ the complement of the theta-divisor ‚ Z D ‚.X I L; d / \ JZ in JZ . Then Q i / of the sheaf H Q i in (2.30) is constant along J , for all i 1. 1) The rank rk.H Z Q l ˝ O .0/ is a trivial subbundle of FQ ˝ O .0/ D H 0 .OZ / ˝ O .0/ . 2) H J J J .0/
Z
Z
Z
Proof. The first assertion is a restatement of Claim 2.6. To see the second assertion .0/ we take two distinct points Œ˛ and Œˇ in JZ and we go back to the identity ˛ Q Q Œ˛/ H.ŒZ; Œˇ/ D H.ŒZ; ˇ in (2.38). Write ˛ 1 1 D ˇ D ; ˇ 1Ct ˛
where t D
ˇ ˛
Q 1 is in H.ŒZ; Œ˛/. This gives the identity Q H.ŒZ; Œˇ/ D
1 Q H.ŒZ; Œ˛/ : 1Ct
I. Reider, Nonabelian Jacobian of Projective Surfaces, Lecture Notes in Mathematics 2072, DOI 10.1007/978-3-642-35662-9 3, © Springer-Verlag Berlin Heidelberg 2013
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Q 3 Some Properties of the Filtration H
34
Q Hence every element h 2 H.ŒZ; Œˇ/ can be written in the form hD
1 s; 1Ct
(3.1)
Q for some s 2 H.ŒZ; Œ˛/. In particular, for ˇ in a small neighborhood1 of ˛, we can expand (3.1) in a convergent power series hD
1 X
t ns ;
nD0
Q l .ŒZ; Œ˛/, for all n 0. This implies where the terms of the series are in H Q Q Q l .ŒZ; Œ˛/ is closed under the that H.ŒZ; Œˇ/ Hl .ŒZ; Œ˛/. Since H 0 multiplication in H .OZ /, we obtain an inclusion Q l .ŒZ; Œ˛/ : Q l .ŒZ; Œˇ/ H H By the first part of the lemma the dimensions of the two vector spaces are equal. This yields an equality Q l .ŒZ; Œˇ/ D H Q l .ŒZ; Œ˛/ ; H
(3.2) .0/
for all Œˇ in a small open neighborhood of Œ˛. Since JZ is path connected, it .0/ u t follows that the equality (3.2) holds for all Œˇ 2 JZ . Q l .ŒZ; Œ˛/ of H 0 .OZ / has the following geometric Remark 3.2. The subring H meaning. Recall the morphism .ŒZ; Œ˛/ W Z
Q P.H.ŒZ; Œ˛/ /
Q l .ŒZ; Œ˛/ in (2.32) and let Z 0 .˛/ be the image of .ŒZ; Œ˛/. Then the space H 0 is isomorphic to H .OZ 0 .˛/ / with the isomorphism given by the pullback by .ŒZ; Œ˛/. More precisely, we have Q l .ŒZ; Œ˛// Z 0 .˛/ D Spec.H and
..ŒZ; Œ˛// W H 0 .OZ 0 .˛/ /
is an isomorphism.
1
In the complex topology of ExtZ1 .
Q l .ŒZ; Œ˛/ H
(3.3)
Q 3 Some Properties of the Filtration H
35
From (3.3) and Lemma 3.1, (2), it also follows that the scheme Z 0 .˛/ is .0/ independent of Œ˛ 2 JZ . In the sequel it will be denoted by Z 0 . Corollary 3.3. Let .0/ be as in 2.7 and let Z .0/ be the universal subscheme over .0/ . There exists a subsheaf F 0 of F ˝ O .0/ such that Q l ˝ O .0/ : F 0 D H J
(3.4)
Furthermore, F 0 is a subsheaf of subrings of F ˝ O .0/ and one has the following factorization f
Z .0/
Z0 .0/
(3.5)
p20
p2
.0/
where Z0 .0/ D Spec.F 0 / and Z0 .0/
f W Z .0/
is the morphism corresponding to the inclusion of sheaves of rings F 0 ,! F ˝O .0/ .
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