  In the present paper we investigate a question stemming from a long-standing
conjecture of Vasconcelos: given a generically a complete intersection perfect
ideal I in a regular local ring R, is it true that if I/I^2 (or R/I^2) is
Cohen-Macaulay then R/I is Gorenstein? Huneke and Ulrich, Minh and Trung, Trung
and Tuan and - very recently - Rinaldo Terai and Yoshida, already considered
this question and gave a positive answer for special classes of ideals. We give
a positive answer for some classes of ideals, however, we also exhibit prime
ideals in regular local rings and homogeneous level ideals in polynomial rings
showing that in general the answer is negative. The homogeneous examples have
been found thanks to the help of J. C. Migliore. Furthermore, the
counterexamples show the sharpness of our main result. As a by-product, we
exhibit several classes of Cohen-Macaulay ideals whose square is not
Cohen-Macaulay. Our methods work both in the homogeneous and in the local
settings.
