Given a smooth affine scheme $X = \mathbb{V}(g)$ over a field of characteristic 0, let $f:X \to \mathbb{A}^1$ be a morphism of schemes. Then, the critical locus is given by $\pi_*(dg \cap df)$ for $\pi:T^*X \to X$. Since $f$ is arbitrary, the intersection locus may have a highly nontrivial/complicated description. Are there algebro/derived geometric tools to describe

- The hessian matrix and index of $f$
- The poincare polynomial of $f$