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0. Introduction 425 1. The quotient construction 428 2. The fiberwise G.I.T. problem 433 3. Cohomology bounds 435 4. Slope-unstable, torsion free sheaves 437 5. Special, torsion bounded sheaves 440 6. Slope-semistable, torsion free sheaves 443 7. Two results in geometric invariant theory 449 8. The construction of Ug(e, r) 454 9. Basic properties of Ug(e, r) 458 10. The isomorphism between Ug(e, 1) and Pg,e 464 References 470

A Compactification over \overline{𝑀_{𝑔}} of the Universal Moduli Space of Slope-Semistable Vector Bundles