Timeline for Number of rational points of a singular cubic surface over a finite field

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Nov 12, 2017 at 17:33	comment	added	Martin Bright		Resolving singularities gives a map $\pi \colon Y \to X$ where $Y$ is a generalised del Pezzo surface. Since your singularities aren't defined over the base field, I guess $\pi$ will give a bijection on $\mathbb{F}_q$-points. You can count points on $Y$ using the Weil conjectures, which in this case comes down to $q^2+1+q$(trace of Frobenius on $\mathrm{Pic}\ Y$). You have $\mathrm{Pic}\ Y \cong \mathbb{Z}^7$ and the Galois action depends on whether your singularities are $A_1$ or $A_2$ and the Galois action on the lines on $X$.
Nov 12, 2017 at 15:15	comment	added	Jason Starr		There is at least one $\mathbb{F}_q$-rational point by Chevalley-Warning. If this point is contained in no line contained in the hypersurface, the tangent planar cubic at this point contains $q$ points, resp. $q+1$ points depending on whether it is nodal, resp. cuspidal. You can iterate the construction of the tangent planar cubic with each of those points.
Nov 12, 2017 at 14:35	history	edited	Hidegol	CC BY-SA 3.0	added 18 characters in body
Nov 12, 2017 at 14:13	comment	added	Jason Starr		The number $N_{\mathbb{F}_q}$ of $\mathbb{F}_q$-rational points of your cubic surface is congruent to $1$ modulo $q$ by the Chevalley-Warning theorem.
Nov 12, 2017 at 7:09	review	First posts
Nov 12, 2017 at 7:27
Nov 12, 2017 at 7:07	history	asked	Hidegol	CC BY-SA 3.0