What is the shape of the zeta function of a singular hypersurface? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T11:27:35Z http://mathoverflow.net/feeds/question/82224 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82224/what-is-the-shape-of-the-zeta-function-of-a-singular-hypersurface What is the shape of the zeta function of a singular hypersurface? Hugo Chapdelaine 2011-11-29T22:15:38Z 2011-11-30T14:33:52Z <p>So let $X$ be a projective hypersurface inside $\mathbb{P}_{\mathbb{Z}}^n$ of degree $d$. Assume that</p> <p>(a) $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are irreducible,</p> <p>(b) and that $X(\mathbb{C})$ and $X(\overline{\mathbb{F}}_p)$ are smooth varieties (therefore of Zariski dimension $n-1$).</p> <p>Then we know from the classical Weil conjectures and (Hard Lefschetz) that <code> $$Z(X/\mathbb{F}_p,T)=\frac{P_{n-1}(T)^{(-1)^n}}{(1-T)(1-pT)\cdots (1-p^{n-1}T)},$$ </code> where $$P_{n-1}(T)=\prod_{j=1}^{b_{n-1}}(1-\alpha_{j} T),$$ with <code>$|\alpha_{j}|=p^{(n-1)/2}$</code> where the $(n-1)$-th Betti number of $X(\mathbb{C})$ is given explicitly by <code> $$b_{n-1}=\frac{(d-1)^{n+1}+(-1)^{n+1}(d-1)}{d}.$$ </code> The last formula being a direct consequence of Gauss-Bonnet.</p> <p><strong>Q</strong>: So for a general connected projective hypersurface $X$ of $\mathbb{P}_{\mathbb{Z}}^n$ such that</p> <p>(i) <code>$\dim_{\overline{\mathbb{F}}_p}(X(\overline{\mathbb{F}}_p))=\dim_{\mathbb{C}}(X(\mathbb{C}))=n-1$</code>,</p> <p><strong>which is no more assumed to be smooth (over $\overline{\mathbb{F}}_p$ and $\mathbb{C}$)</strong>, what is the "shape" of $Z(X/\mathbb{F}_p,T)$?</p> <p>I guess that a precise answer to this question should involve a description of the singular locus of $X(\overline{\mathbb{F}}_p)$ and $X(\mathbb{C})$ (the number of components of the singular locus and the type of singularity for each intersection of two components).</p> <p>P.S. Note that if one has a precise recipe for the zeta function of such a hypersurface, then by the inclusion-exclusion principle one gets a description of the zeta function of a general (equi-dimensional) projective scheme $X$ of finite type over $Spec(\mathbb{Z})$. </p> <p><strong>added</strong>: By shape I mean an explicit factorization of the numerator and denominator of $Z(T)$ that reflects the geometry of $X(\mathbb{C})$. I know nothing about intersection homology but I guess that the shape of the zeta function should encode some data about $IH^{*}(X)$. For example, in the smooth case, the number of reciprocal roots (or poles depending on the parity of $i$) with complex absolute value $p^{i/2}$ of $Z(T)$ equals the $i$-th Betti number of $X(\mathbb{C})$.</p>