If $X\to Y$ is a finite map of connected proper algebraic curves over a field, then for any point $y\in Y$, the sum $\sum e_xf_x=n$ of ramification times inertia degrees over points $x$ mapping to $y$ equals the degree of the map.

Now if $X\to Y$ is any finite flat map of connected proper algebraic varieties, the Hilbert polynomial $$H(X_y)\ =\ n$$ of the fibres will be invariant since the map is flat, and constant since it's finite.

The question: Does $H(X_y)$ have an expression like $\sum e_xf_x$ as a sum of products of functions of $x$, each of which has a satisfying interpretation?

When $X,Y$ are algebraic surfaces and $y$ is a curve (i.e. its generic point) or a closed point, how should they be visualised?

Of course this boils down to, for a map of local rings $(B,\mathfrak{m}_B)\to(A,\mathfrak{m}_A)$, computing $\dim_{A/\mathfrak{m}_A}B/\mathfrak{m}_A$ in terms of ``geometric'' data. But this might not be the best way of going about it; it was suggested to me that the Grothendieck Riemann Roch formula applied to $f_*\mathcal{O}_X$ might give an answer. I haven't been able to find anything this way yet.

  • $\begingroup$ I wasn't quite sure whether MSE or MO was more appropriate. I'd be happy for it to be migrated if I made the wrong choice. $\endgroup$ – Meow Mar 12 at 21:46

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