I would like to know whether the $\ell$-adic intersection cohomology on the étale site of a projective nodal cubic over a finite field is trivial in degree $1$ (i.e., whether or not the first betti number is zero), and if not I would like to have some idea what it looks like. For concreteness one could restrict for example to the case of $\textrm{Proj}(\mathbb{F}_p[x,y,z]/\langle zy^2-x^3-zx^2\rangle)$ for some prime $p$. I don't have any concrete evidence why it should be trivial, but I'll try to explain why I suspect it is.

Topologically, a nodal cubic over the complex numbers looks like a sphere identified at two points. Using e.g. a Mayer-Vietoris sequence, it's easy to compute the singular intersection cohomology of such a surface, and it turns out to be trivial in degree $1$.

Now for, e.g., a smooth projective variety $X$ over $\textrm{Spec}(\mathbb{Z})$, one can use base change and Artin's comparison theorem to show that the $\ell$-adic (étale) betti numbers of $X$'s reduction mod $p$ are the same as the singular cohomology betti numbers of $X\times_{\mathbb{Z}}\textrm{Spec}(\mathbb{C})$ with its analytic topology.

My understanding of $\ell$-adic intersection cohomology is (at best) extremely vague, so I don't know if similar results can be used to relate the $\ell$-adic intersection cohomology of a nodal cubic over a finite field to the singular intersection cohomology of its analytification.

However, (I don't know any French so I may be misunderstanding) in trying to skim through the introduction to BBD it appears as though in section $4$ an analogue of Artin's theorem is proved, and in section $6$ the authors develop mechanisms for comparing the intersection cohomology of a scheme over the complex numbers to its cohomology over a finite field. Therefore it seems plausible that the degree-$1$ $\ell$-adic intersection cohomology of $\textrm{Proj}(\mathbb{F}_p[x,y,z]/\langle zy^2-x^3-zx^2\rangle)$ should also be trivial.


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