Let $C$ be a reducible nodal curve over complex numbers with two smooth components $C_1$ and $C_2$ intersecting at the only node $P$. Let $E$ be a $\omega$ semistable vector bundle over $C$ of rank $r$ and Euler Characteristic $\chi$, where $\omega=(\omega_1,\omega_2)$ is a polarization. Let $E_1$ and $E_2$ be the restrictions of $E$ to the components $C_1$ and $C_2$ with Euler Characteristics $\chi_1, \chi_2$ respectively. Then, Teixidor has shown (in a more general setting) in "Moduli spaces of (semi)stable vector bundles on tree-like curves, Math. Ann. 209, (1991), 341-348. " that $$\omega_1\chi\leq \chi_1\leq \omega_1\chi+r$$ $$\omega_2\chi\leq \chi_2\leq \omega_2\chi+r$$ Further, she has shown that, if $E$ is any vector bundle on $C$ such that $E_i's$ are semistable on $C_i's$ and the above inequalities are true then $E$ is $\omega$ semistable.
Question: Hence, it is natural to ask when the converse is true. That is, for $E$ a $\omega$-semistable vector bundle, can we find conditions (mostly on the Euler Characteristics of $E$ and its restrictions) such that $E_i's$ are semistable?
In the case of rank 2, Nagraj and Seshadri have answered this question in "Degenerations of the moduli space of vector bundles on curves I, Proc. Indian Acad. Sci. (math. Sci.), 107, (1997), pp.101-137 " Theorem 5.1: If $w_1\chi\notin \mathbb{Z}$, $E$ is $\omega$-semistable and $\chi_1$, $\chi_2$ satisfy the inequalities $\omega_1\chi<\chi_1<\omega_1\chi+1$ and $\omega_2\chi+1<\chi_2<\omega_2\chi+2$, then $E_1,E_2$ are semistable.
My Question
- Can we find an example of a $\omega$ semistable vector bundle $E$ on $C$ of rank $r>2$ such that at least one of the $E_i's$ is not semistable?
- For $r=2$, if we drop the condition $\omega_1\chi\notin\mathbb{Z}$, then can we find an example of a $\omega$ semistable vector bundle $E$ on $C$ such that at least one of the $E_i's$ is not semistable?
