The HFEv- signature scheme is one of the most studied multivariate schemes and one of the major candidates for the upcoming standardization of post-quantum digital signature schemes. In this paper, we propose three new attack strategies against HFEv-, each of them using the idea of projection. Especially our third attack is very effective and is, for some parameter sets, the most efficient known attack against HFEv-. Furthermore, our attack requires much less memory than direct and rank attacks. By our work, we therefore give new insights in the security of the HFEv- signature scheme and restrictions for the parameter choice of a possible future standardized HFEv- instance.