Transform of the tautological sheaf of a line bundle

Corollary 4.3.7 In characteristic0,σ^∗L^[n]is poly-stable. In any characteristic, it is semistable.

Proof.

The first assertion follows from remark 4.3.1. To prove the second assertion, we use the fact that a maximal semistable subsheaf isS_n-linearized, and then follow the proof of proposition 4.3.6.

4.3.3 Transform of the tautological sheaf of a line bundle

In this paragraph we want to use the proof of Ein and Lazarsfeld of the stability ofMLfor line bundles (cf. [EL92]), to show that when degL>2g+nthe total transform

NL =M_L[n] =ker(H⁰(C,L)⊗ O_SnCL^[n]) is stable.

In [EL92] the stability of the Picard bundle is shown proving the stability of a total transform, we recall their argument, as it is useful for a better under-standing of the stability ofNL.

4.3. SYMMETRIC PRODUCT OF A CURVE 59 Consider a pointed smooth curve(C,x0)of genus g > 1, for alldthere is a Picard scheme Pic^d(C), which is a fine moduli scheme for line bundles of degreedonC. And a universal sheafU onC×Pic^d(C)such thatU_|C×ξ = L_ξ for all points ξ ∈ Pic^d(C), representing a line bundle L_ξ, andU_|x

0×Pic^d(C) = O_Picd(C).

The Picard sheafPdis the push forward ofUto Pic^d(C). Whend>2g−1, Pdis a vector bundle, whose fiber upon a pointξ∈Pic^d(C)isH⁰(C,L_ξ).

The Picard schemes are all isomorphic, an isomorphism Pic^d(C)−→˜ Pic^d+1(C) is given by the mapL7→L⊗ O_C(x0). In degree 0 the Picard scheme is the Jaco-bian of the curveH¹(C,O_C)/H¹(C,Z), and is smooth projective of dimension g.

The Picard schemes Pic^d(C)have a canonical polarization. It is given by the theta divisorΘ⊂ Pic^d(C), which is the ample divisor represented by the image of the map

C^g−1→Pic^g−1(C),(x₁, . . . ,xg−1)7→ O_C(x₁+· · ·+xg−1)

in Pic^g−1(C). By the isomorphisms above it corresponds to ample divisors on all Picard schemes.

Given a line bundleA∈Pic^d(C), we can consider the applications v^r_A:C^r→Pic^d−r(C), (x₁, . . . ,xr)7→A⊗ O_C(−x₁− · · · −xr). Then by Poincar´e’s formula, whenr<gthe class of the image is

Wr := [Θ]^g−r/(g−r)!

in particular, the class of the imagev¹_A(C)is[Θ]^g−1/(g−1)!.

Hence to prove stability of P_d with respect toΘ, i.e. to show that for all subsheavesF,→P_dwe have

c₁(F).[_Θ]^g−1

rk(F) < ^c1(P_d).[_Θ]^g−1 rk(Pd) ^,

it is sufficient to show that(v¹_A)^∗P_dis stable for anA∈Pic^d+1(C), and then ap-ply theorem 1.2.25. And this is done, whend>2g, by showing that(v¹_A)^∗P_d= MA⊗ O_C(x0), and that the transformMAis stable.

Remark 4.3.8 To show that the stability of Pdis implied by the stability of a curve whose class is a multiple of [Θ]^g−1, it is not used in [EL92] a restric-tion result like theorem 1.2.25 but rather the generality of the line bundleA∈ Pic^d+1(C)_.

Remark 4.3.9 Actually Ein and Lazarsfeld show something stronger than sta-bility ofML, they prove thatMLis cohomologically stable, but we will not use such concept.

We can use the same argument to show the stability of the total transform NLof a tautological bundleL^[n]onSⁿC.

Theorem 4.3.10 Let C be a smooth curve of genus g >2, and let L be a line bundle on C of degree d > 2g+n. Then NL is a stable bundle on SⁿC with respect to the polarizationH.˜

To prove this we show that for all (x1, . . . ,xn−1) ∈ Cⁿ⁻¹ the restriction NL|x₁+···+x_n−1+Cis stable, and then use theorem 1.2.25

Proof.

We want to show that for all(x1, . . . ,xn−1)∈Cⁿ⁻¹the restrictionN_{L|x1+···+xn−1+C} is stable.

To prove it we could just use the restriction to x1+· · ·+xn−1+C of the exact sequence

0→π1∗(π₂^∗L⊗ O(−Z))→H⁰(C,L)⊗ O_S2C→L^[n]→0 ,

but it seems more clarifying if we look at our total transform as the pull back of the Picard bundle via by the map

vⁿ_L :SⁿC→Pic^d−n(C), (x₁, . . . ,xn)7→L⊗ O_C(−x₁− · · · −xn). Claim: (vⁿ_L)^∗(P_d−n) =Nf_L⊗ O_Sⁿ_C(x₀+Sⁿ⁻¹C)_.

From the claim we deduce thatN_{L|x1+···+xn−1+C}isM_{L(−x1−···−xn−1)}tensorized by a line bundle, hence it is stable by Ein and Lazarsfeld results, or by Butler’s theorem (3.1.4).

To prove the claim, we can observe that(vⁿ_L)^∗(P_d−n) = π1∗((vⁿ_L×1_C)^∗U) by the theorem on cohomology and base change. Then use see-saw principle to show that

(vⁿ_L×1C)^∗U =π^∗₂L⊗ O_SnC×C(−Z)⊗π^∗₁O_SnC(x0+Sⁿ⁻¹C),

where π₁ andπ₂are the two naturale projections of SⁿC×C, p1 is the pro-jection of Pic^d−n(C)×C to the first factor. And U is the universal sheaf on Pic^d−n(C)×C.

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