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Restricted tangent bundle on space curves. (English) Zbl 0859.14011
Teicher, Mina (ed.), Proceedings of the Hirzebruch 65 conference on algebraic geometry, Bar-Ilan University, Ramat Gan, Israel, May 2-7, 1993. Ramat-Gan: Bar-Ilan University, Isr. Math. Conf. Proc. 9, 283-294 (1996).
Let $$H(n,d,g)$$, $$g\geq 1$$, be the variety of smooth connected curves of genus $$g$$ and degree $$d$$ embedded in $$\mathbb{P}^n$$. For $$X\in H(n,d,g)$$, let $$R_X$$ denote the restriction of the tangent bundle of $$\mathbb{P}^n$$ to $$X$$. In this paper, the authors study semistability and simplicity of $$R_X$$.
Theorem 1: (Bogomolov) For $$n=2$$, $$R_X$$ is stable if $$d\geq 3$$ ($$g\geq 1$$). The splitting type of $$R_X$$ is (3,3) if $$X$$ is a conic and $$(2,1)$$ if $$X$$ is a line in $$\mathbb{P}^2$$.
Theorem 2: For $$n=3$$, $$d\geq g+3$$, there exists a nonempty dense open subset of $$H(3,d,g)$$ consisting of $$X$$ with $$R_X$$ semistable and moreover, simple if $$g\geq 2$$.
Theorem 2 is proved by induction on $$(d,g)$$ using degenerations of smooth curves $$X$$ to reducible reduced curves $$Y$$ with ordinary double points. For this purpose the notions of Harder-Narasimhan polygons and strata are generalized to vector bundles of $$Y$$.
J. Simonis [Math. Ann. 192, 262-278 (1971)] and D. Laksov [Astérisque 87/88, 207-219 (1981; Zbl 0489.14008)] had independently proved that if $$X$$ is a closed subvariety of $$\mathbb{P}_n$$ which is nonsingular in codimension 1 and linearly normal, then $$R_X$$ is decomposable if and only if $$X$$ is a rational curve. – For rational curves, $$R_X$$ was studied by L. Ramella [Thesis (Nice 1988)] and F. Ghione, A. Iarrobino and G. Sacchiero [preprint (1988)].
For the entire collection see [Zbl 0828.00035].

##### MSC:
 14H50 Plane and space curves 14H60 Vector bundles on curves and their moduli 14H10 Families, moduli of curves (algebraic) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
##### Keywords:
space curves; tangent bundle