  Let $M$ be the moduli space of generalized parabolic bundles (GPBs) of rank
$r$ and degree $d$ on a smooth curve $X$. Let $M_{\bar L}$ be the closure of
its subset consisting of GPBs with fixed determinant ${\bar L}$. We define a
moduli functor for which $M_{\bar L}$ is the coarse moduli scheme. Using the
correspondence between GPBs on $X$ and torsion-free sheaves on a nodal curve
$Y$ of which $X$ is a desingularization, we show that $M_{\bar L}$ can be
regarded as the compactified moduli scheme of vector bundles on $Y$ with fixed
determinant. We get a natural scheme structure on the closure of the subset
consisting of torsion-free sheaves with a fixed determinant in the moduli space
of torsion-free sheaves on $Y$. The relation to Seshadri--Nagaraj conjecture is
studied.
