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Consider $X$ a smooth cubic surface in $\mathbb{P}^3$, and let $l_1,...,l_6$ be six disjoint lines contained in $X$. What is the linear system giving the blow-down map $X \to \mathbb{P}^2$, so that th …

For a surface $S$, supersingular means that the étale cohomology group $H^{2}(S,\mathbb{Q}_\ell)$ ($\ell$ a prime, prime to the characteristic $p$) is generated by divisors on $S$ (thus the Picard num …

Let $S$ be a smooth surface defined over a finite field $K$ of char. $p$. Let $G$ be a finite group of automorphisms of $S$. Let $Z\to S/G$ be the minimal resolution of the quotient of $S$ by $G$. Su …

I am looking for a reference that describes what are the possible singularities of a curve $C=C_1 \cup C_2$ which is the union of two smooth irreducible curves $C_1,\,C_2$ on a smooth surface. Doing …

I figured out what @Mohan told me and wrote a proof Let $C_{1},\,C_{2}$ be two smooth curves on a smooth surface and let $p$ be an intersection point of $C_{1}$ and $C_{2}$. One can suppose that ther …

Let $A$ be an Abelian variety defined over the finite field with $q$ elements. Let $P_i(T)$ be the characteristic polynomial of the action of the Frobenius on the $i^{th}$ étale cohomology group. Is …