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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

  1. The hessian matrix and index of $f$
  2. The poincare polynomial of $f$
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  • $\begingroup$ you mean the index of a critical point of $f$? $\endgroup$
    – Elden Elmanto
    Commented Sep 6, 2016 at 15:13
  • $\begingroup$ The hessian can be defined in a completely algebraic manner at a singular point ($df=0$). Without loss of generality assume $f=0$ at the critical point. Consider the 2nd order jet sequence $0 \to Sym^2 \mathcal{T}^*_X \to \mathcal{J}^2 \to \mathcal{J}^1 \to 0$. Since the 1-st order jet vanishes at the critical point $f$ defines a unique symmteric bilinear form on the tangent space. Regarding the second part it depends what you mean by "poincare polynomial. There are many cohomology theories for smooth affine schemes. $\endgroup$
    – Saal Hardali
    Commented Jul 22, 2017 at 12:12

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