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].

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].

Reviewer: U.N.Bhosle (Bombay)