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.