Seshadri constants of the anticanonical divisors of Fano manifolds with large index

Seshadri constants, introduced by Demailly, measure the local positivity of a nef divisor at a point. In this paper, we compute the Seshadri constants of the anticanonical divisors of Fano manifolds with coindex at most $3$ at a very general point. As a consequence, if $X$ is a nonsingular Fano threefold which is very general in its deformation family, then $\epsilon (X,-K_X;x)\le 1$ for all points $x\in X$ if and only if $\mathrm{\mid }-K_X\mathrm{\mid }$ is not base point free.