  In this paper we study the local zero behavior of orthogonal polynomials
around an algebraic singularity, that is, when the measure of orthogonality is
supported on $ [-1,1] $ and behaves like $ h(x)|x - x_0|^\lambda dx $ for some
$ x_0 \in (-1,1) $, where $ h(x) $ is strictly positive and analytic. We shall
sharpen the theorem of Yoram Last and Barry Simon and show that the so-called
fine zero spacing (which is known for $ \lambda = 0$) unravels in the general
case, and the asymptotic behavior of neighbouring zeros around the singularity
can be described with the zeros of the function $ c J_{\frac{\lambda -
1}{2}}(x) + d J_{\frac{\lambda + 1}{2}}(x) $, where $ J_a(x) $ denotes the
Bessel function of the first kind and order $ a $. Moreover, using
Sturm-Liouville theory, we study the behavior of this linear combination of
Bessel functions, thus providing estimates for the zeros in question.
