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 times 3) permanent is 7. We also prove that for (n> 3), there is no nonsingular hypersurface in ({mathbb {P}}^n) of degree d that has an expression as a determinant of a (d times d) matrix of linear forms, while on the other hand for (n le 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.