Mukai varieties as hyperplane sections

Let (M,L) be a smooth (n+1)-dimensional variety polarized by an ample and spanned line bundle L. Let A be a smooth member of t the linear system defined by L. Assume that $n> 4$ and that (A,H_A) is a Mukai variety, i.e., -K_A=(n-2)H_A for some ample line bundle H_A on A. Let H be the line bundle on M which extends H_A. We show that M is a Fano variety and either H is ample, in which case the cones of effective 1-cycles NE(A) and NE(M) on A and M coincide, or $\grk(H)=n$, H is semiample and (M,L) has a structure of a conic fibration. Then most of the paper is devoted to classify the pair(M,L) in the case whenL is very ample.