# Some queries on Moduli Space of Parabolic Vector Bundles on curve

In "Moduli of Vector Bundles on curves with Parabolic Structures"-Bulletin of the American Mathematical Society Volume 83, Number 1, January 1977 the author announces the following result on moduli space of parabolic vector bundles on curves-

**Let $VB(d,\alpha)$ denote the category of parabolic semistable vector bundles $V$ on a smooth projective curve $X$ with a single point $P\in X$ of fixed weight $\alpha=(\alpha_1,\alpha_2, \cdots \alpha_n)$ and fixed ordinary degree $d$. Also for each $V\in S$ $gr(V)$ is defined to be $\oplus V_i/V_{i-1}$ where $V=V_n\supset V_{n-1}\supset V_{n-2}\cdots \supset V_{0}=0$ is a filtaration such that each $V_i$ is parabolic sub-bundle of $V$ and each $V_i/V_{i-1}$ is a stable parabolic bundle of degree 0. Define $V$ and $V'$ to be equivalent if $grV=grV'$. Let $M(d,\alpha)$ be the set of equivalence classes under this relation. **

Theorem- Suppose that $g=$genus of $X\geq 2$. Then there is a natural structure of a normal projective variety on $M(d,\alpha)$ of dimension $n^2(g-1)+\delta$ where $\delta$ is the dimension of the flag variety of flags in an $n$-dimensional vector space of type given by the type of of the underlying quasi-parabolic structure. Further $M(d,\alpha)$ is smooth at the points $V$ where $V$ is stable.

In "Moduli of Vector Bundles on curves with parabolic structure"-Math Ann 248, 205-239(1980)- Mehta and Seshadri prove the theorem-

P-225 (4. Existence of the Moduli Space)

Let X be a smooth projective curve of genus $g\geq 2$ over an algebraically closed field $\mathcal{k}$. Consider the set of all parabolic semi-stable bundles of rank $k$, fixed quasi-parabolic structure at a given point $P$, fixed weights $0<\alpha_1<\alpha_2\cdots\alpha<1$ with all $(\alpha_i)$ rational, fixed degree $d$ and parabolic degree $0$. Denote this set by $S(k,\alpha,d,0)$ or simply $S$.Define $V$ and $V'$ to be equivalent if $grV=grV'$

Theorem 4.1 1) On the set of equivalence classes of S, there exists a natural structure of a normal projective variety of dimension $k^2(g-1)+1+dim \mathcal {F}$, where $\mathcal {F}$ is the flag variety of type determined by the quasi-parabolic structure at $P\in X$

Questions

1. In the first paper the ordinary degree $d$ of the vector bundle could be anything, whereas in the second paper they have put a constraint $\operatorname{ParDeg}(V)=0$ which is same is saying $d=-\sum m_i \alpha _i$, where $m_i$'s are the multiplicities. Are these constraints necessary for the construction of the moduli space?

2. In the first paper there was no constraint on parabolic weights, whereas in the second paper parabolic weights are chosen from rational numbers. Is the construction of moduli space still valid if we take $(\alpha_i)$ irrational?

3. The dimensions of the moduli space are not equal in these two papers. Am I missing something?