2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/70321Let $Y$ be a smooth projective curve of genus $g\ge 2$ and let $M_{r,d}(Y)$ be the moduli space of stable vector bundles of rank $r$ and degree $d$ on $Y$. A classical conjecture of Newstead and Ramanan states that $ c_i(M_{2,1}(Y))=0$ for $i>2(g-1)$ i.e. the top $2g-1$ Chern classes vanish. The purpose of this paper is to generalize this vanishing result to the rank 3 case by generalizing Gieseker's degeneration method. More precisely, we prove that $c_i(M_{3,1}(Y))=0$ for $i>6g-5$. In other words, the top $3g-3$ Chern classes vanish. Notice that we also have $c_i(M_{3,2}(Y))=0$ for $i>6g-5$.Algebraic Geometry14H60; 14F25; 14F42text