Semistable sheaves on projective varieties and their restriction to curves
Semistable sheaves on projective varieties and their restriction to curves
复制标题
射影簇上的半稳定滑轮及其对曲线的限制
DOI：
10.1007/bf01450677
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发表时间：
1982
影响因子：
1.4
通讯作者：
A. Ramanathan
中科院分区：
文献类型：
--
作者：
V. Mehta;A. Ramanathan
令 X 为代数闭域 k 上维度为 n 的非奇异射影簇。设 H 是 X 上的一个非常充足的线束。如果 V 是 X 上的无扭相干束，我们将 deg V 定义为 cl(V)、c~(H)"~ 和/~(V) = deg V/rk V。如果对于 V 的所有真子滑轮 W，我们有 #(W) < #(V) [resp. #(W) < #(V)]（参见 [7, 14]）在本文中，我们证明如果 V 在 X 上是半稳定的，那么它对足够高阶的一般完全相交曲线的限制是半稳定的（定理 6.1）。为了给出证明的想法，假设 X 是一个曲面，V 是一个阶为 2 的向量丛。当且仅当它在 IPH~ H" 函数域上定义的通用曲线 Ym 上不是半稳定时，V 对 m 阶的一般曲线 C" 的限制才不是半稳定的。令/S," 为 Y, 上的线丛，与 VI Y,, 的半稳定性相矛盾（参见第 4.1 和 4.2 节）。首先我们证明 L,, 唯一地扩展到 X 上的线丛 L m （命题 2.1）。如果我们能得到 L" 作为 V 的子丛，那么 L" 就会与 V 的半稳定性相矛盾。所以我们希望限制映射 H~ Hom(Lm, V))~H~ Hom(L m, V)) 是满射的。现在，对于固定的 L，从 Enriques-Severi 的引理（命题 3.2；[6，推论 7.8]）得出，H~ Horn(L, V))~H~ Horn(L, V)) 对于大 m 是满射。因此，如果 L" 在无限多个 m 内保持相同的线丛 L 就足够了。为了证明 L,, = L，我们构建了一个退化曲线族 D f ~S, X x S 3 D p ~X，使得通用纤维是 2"+1 次的曲线 Ct"+ 1)，特殊纤维是具有 2" 次非奇异分量 CI") 的简化曲线（参见第 5 节）。令 (m) 表示2”。将子束 Lt,,+~)[Ct"+~) 扩展到 D 上的 p*(V) 子束并将扩展限制为 CI") 给出了 V[CIm ) 的线子束最大次数的下界 (命题 4.3)。这意味着 degL,, 是有界的（引理 6.5.1），因此对于 m 的无限子序列，degL,, 是常数。如果 degLtm + r)= degLt,,) 通过用退化族改进上述论证，可以证明 Lt"+,)[ CI")
Let X be a nonsingular projective variety of dimension n over an algebraically closed field k. Let H be a very ample line bundle on X. If V is a torsion free coherent sheaf on X we define deg V to be cl(V), c~(H)"~ and/~(V) = deg V/rk V. We call V sernistable (resp. stable) if for all proper subsheaves W of V we have #(W) < #(V) [resp. #(W) < #(V)] (cf. [7, 14]). In this paper we prove that if V is semistable on X then its restriction to a general complete intersection curve of sufficiently high degree is semistable (Theorem 6.1). To give an idea of the proof assume X is a surface and V a vector bundle of rank 2. The restriction of V to a general curve C" of degree m is not semistable if and only if it is not semistable on the generic curve Ym defined over the function field of IPH~ H"). Let/S," be the line bundle on Y,, contradicting the semistability of VI Y,, (cf. Sects. 4.1 and 4.2). First we show that L,, extends uniquely to a line bundle L m on X (Proposition 2.1). If we can get L" as a subbundle of V we are through, for then L" would contradict the semistability of V. So we would like the restriction map H~ Hom(Lm, V))~H~ Hom(L m, V)) to be surjective. Now for fixed L it follows from the lemma of Enriques-Severi (Proposition 3.2; [6, Corollary 7.8]) that H~ Horn(L, V))~H~ Horn(L, V)) is surjective for large m. Therefore it is enough if the L" remain the same line bundle L for infinitely many m. To prove that L,, = L we construct a degenerating family of curves D f ~S, X x S 3 D p ~X, such that the generic fibre is a curve Ct"+ 1) of degree 2 "+ 1 and the special fibre is a reduced curve with two nonsingular components CI") of degree 2" (cf. Sect. 5). Let (m) denote 2". Extending the subbundle Lt,,+~)[Ct"+~) to a subsheaf of p*(V) on D and restricting the extension to CI") gives a lower bound for the maximal degree of a line subbundle of V[CIm ) in terms of that for V[Ctm+I ) (Proposition 4.3). This implies that degL,, is bounded (Lemma 6.5.1) so that for an infinite subsequence of m, degL,, is constant. If degLtm + r)= degLt,,) by refining the above argument with the degenerating family one can prove that Lt"+,)[ CI")