When is the moduli of generalized parabolic bundles with fixed determinant smooth?

Question

Let $X$ be a smooth, projective curve of genus at least $2$, $x_1, x_2$ two distinct closed points, $d$ an odd integer and $\alpha$ a positive real number less than $1$. By a generalized parabolic bundle of rank $2$ on $X$, we mean a triple $(E,F_1(E),(0,\alpha))$, where $E$ is a rank $2$ vector bundle on $E$ and $F_1(E)$ a two dimensional vector subspace of $E_{x_1} \oplus E_{x_2}$. We know that there exists a moduli space of semi-stable, generalized parabolic bundles on $X$ as mentioned here. This moduli space is smooth if $\alpha$ is close to $1$ (in partcular if stable GPB=semi-stable GPB).

One can define the determinant of a GPB, $\det(E,F_1(E),(0,\alpha)):=\wedge^2 E$. Using this definition, it is known that the moduli of semi-stable generalized parabolic bundles with fixed determinant is irreducible, normal with rational singularities. My question is: Is there any condition under which this moduli of semi-stable generalized parabolic bundles with fixed determinant is smooth?

Answer 1

It seems to me that the definition of generalized parabolic structure given in the question differs from the one given in the following paper (a GPB should fix a flag in the global sections of a twist of the vector bundle):

• U. Bhosle. Generalized parabolic bundles and applications. II. Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 403–420. (link to paper on journal website)

In any case, Section 3 of that paper discusses the determinant of GPBs and Theorem 3 of the paper provides an explicit condition for smoothness of the fixed-determinant moduli space (similar to the $\alpha$ near 1 condition for the full moduli space).