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?



When is the moduli of generalized parabolic bundles with fixed determinant smooth?
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)](https://link.springer.com/article/10.1007%2FBF02837696)


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).


