# Algebraic geometry by Shafarevich I.R.

By Shafarevich I.R.

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L). 28 2 Preliminaries (i+1) (i) (i) (1) (l+1) (i) Let fi : VD −→ VD , ti : VD −→ VD and si : VD −→ VD denote the inclusions. Let us consider the complex C1 (V1∗ , V2∗ ) given as follows: (1) (l+1) Hom V1 , V2 d−1 l+1 (i) −→ (i) Hom V1 , V2 i=1 l d0 (i+1) −→ Hom V1 (i) , V2 i=1 Here, the first term stands in the degree −1. The differentials di are given as follows: d−1 (a) = si ◦ a ◦ ti i = 1, . . , l + 1 d0 (b1 , . . , bl ) = −f1 ◦b1 +b2 ◦f1 , −f2 ◦b2 +b3 ◦f2 , . . 3) More precisely, ϕ0 is the projection induced by the identifications V0 = V (1) and si+1 ◦ ai · ti , and ϕ2 is the identity.

In the oriented case, we use the symbol Mss y, L, (δ, ) as usual. We obtain a stack Mss y, [L], (δ, ) from Mss y, L, (δ, ) as in the case of Mss (y, [L], δ). Then, the fixed point set is as follows: M1 M Gm (I) M2 I∈Dec(m,y,δ) Here, M Gm (I) is isomorphic to a moduli stack of objects (E1 , φ, E2 , ρ, F (1) , F (2) ) with the following properties: • (E1 , φ) is a δ-semistable L-Bradlow pair, and F (1) is a full flag of H 0 (X, E1 (m)) such that (E1 , φ, F (1) ) is δ, k(I) -semistable. • E2 is a semistable torsion-free sheaf, and F (2) is a full flag of H 0 (X, E2 (m)) (2) such that E2 , Fmin(I2 ) is -semistable reduced O(−m)-Bradlow pair, where denotes any sufficiently small positive number.

We set Q E0 · eT0 , E1 · eT1 , . . , Ek · eTk := Q Ei · eTi , Ej · eTj . i