A lower bound for the determinantal complexity of a hypersurface

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a determinant of a d×d matrix of linear forms while on the other hand for n ≤ 3, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.