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. 



How to Interpret the Euler Characteristic of Complex Algebraic Varieties
Hi Dori! I think that you will find Paolo Aluffi's paper ["Computing characteristic classes of projective schemes"](http://arxiv.org/abs/math/0204230) 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"](http://arxiv.org/abs/0807.1690). 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](http://arxiv.org/abs/1005.4470)).


