Determinant morphism for singular varieties

Let $X$ be a projective variety (possibly singular) over an algebraically closed field of any characteristic and $\mathcal{F}$ be a coherent sheaf. In this article, we define the determinant of $\mathcal{F}$ such that it agrees with the classical definition of determinant in the case when $X$ is non-singular. We study how the Hilbert polynomial of the determinant varies in families of singular varieties. Consider a singular family such that every fiber is a normal, projective variety. Unlike in the case when the family is smooth, the Hilbert polynomial of the determinant does not remain constant in singular families. However, we show that it exhibits an upper semi-continuous behaviour. Using this we give a determinant morphism defined over flat families of coherent sheaves. This morphism coincides with the classical determinant morphism in the smooth case. Finally, we give applications of our results to moduli spaces of semi-stable sheaves on $X$ and to Hilbert schemes of curves.