Let $X$ be a projective (or affine) variety over $\mathbb{C}$ defined by some homogenous ideal $I = (f_1,\ldots,f_n)$. How can we interpret the Euler characteristic of $X$ other than as just an invariant to distinguish non isomorphic objects? What I mean is what could it tell us if anything about either $X$ or the polynomials defining $I$ if $\chi(X) > 0$? How about if $\chi(X) = 0$ or $\chi(X) < 0$? How about in the case of $X$ being a hypersurface cut out by the single (homogeneous) polynomial $f$. What can we deduce about $f$ if $\chi(X) > 0, \chi(X) = 0,$ or $\chi(X) < 0$. I've been trying to relate conditions on the Euler characteristic to anything explicit about the polynomials defining $X$ but haven't had any success so anything that could be said would be interesting and helpful.
[Math] How tonterpret the Euler Characteristic of Complex Algebraic Varieties
ag.algebraic-geometry
Best Answer
Hi Dori! I think that you will find Paolo Aluffi's paper "Computing characteristic classes of projective schemes" useful, if not for the exact formulas, at least for the algorithm.
Among other things it proves the version of the formula in J.C. Ottem's answer for singular hypersurfaces. If $f\in \mathbb C[x_0,\dots,x_n]$ is a non-constant homogeneous polynomial, let $X=\mathcal Z(f)$ and $\Sigma = \mathcal Z_{\mathbb P^n}(\partial_0 f,\dots,\partial_n f)$. If we denote by $g_0,g_1,\dots,g_{n-1}$ the degrees of the gradient morphism $$\mathbb P^n\backslash \Sigma\to \mathbb P^n, x=(x_0,\dots,x_n)\to (\partial_0f(x),\dots,\partial_nf(x))$$ then we have $$\chi(X)=n+\sum g_i (-1)^{n+1-i}.$$
As far as restricting to graph hypersurfaces, I would say look at the recent papers of Aluffi and Marcolli but you know more than me about these matters so... :)
Added: A toy application of this is in the case of graphs $\Gamma_n$ which have two vertices and $n$ parallel edges, so called banana graphs. the formula above gives $$\chi(X_{\Gamma_n})=n+(-1)^n$$ so the Euler characteristic determines the graph. The motivic generalization is done in "Feynman motives of banana graphs". The motivic viewpoint also shows that one shouldn't expect a nice relation to hold for all graphs, since graph hypersurfaces are generators for the Grothendieck ring (details and actual statement are here).