# $\ell$-adic intersection cohomology of a nodal cubic over a finite field

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.