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Let $X$ be a smooth projective variety over a finite field. In [Poonen - Bertini theorems over finite fields] it is shown that one can find a smooth geometrically integral hypersurface $S$ of degree …

Gallarati studied contact of surfaces in $\mathbb{P}^3$, that is surfaces $V,W \subset \mathbb{P}^3$ such that $V.W = qD$ with $q$ an integer that is at least 2 and $D$ some curve. I would like to re …

Dear all, Let $f: X \rightarrow Y$ be a morphism of projective varieties over $\mathbb{C}$. Also let $V \subset Y$ be a nontrivial open subvariety and set $U:= f^{-1}(V)$. I would like to show that …

Hi all, on the forum page http://www.groupsrv.com/science/about506648.html one can read the following (i cut out nonimportant parts): Question: Does anyone know any condition (non trivial) that ensu …

My advisor told me the following: Let $\Sigma$ be a singular surface over $\mathbb{C}$ whose singularities are all ordinary quadratic, or more generally Duval singularities. Let $\epsilon: S \rightar …

All varieties will be projective and over $\mathbb{C}$. If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively isomorph …

I was told singular quartic algebraic surfaces in $\mathbb{P}^3_{\mathbb{C}}$ have been completely classified and their singularities have been described. Can anyone provide me with a resource where t …

Today i was talking with my advisor and she told me the following fact: Let $S$ be a singular surface in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$. Writing $\omega_\Sigma$ for the dualizing sheaf and …

The following got no answer on mathstackexchange. I believe it not to be hard, but maybe it is a little specialized? All varieties will be over $\mathbb{C}$ and projective unless stated otherwise. I …

The title explains it all. I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are ch …

Using Jason Starr's comment I was able (I think) to figure out the case of $\pi_1(X)^0$. For anyone who stumbles across this with the same question in mind I add a sketch of the proof in an answer. Fo …

Let $CH_0(X)^0$ denote the group of zero cycles of degree zero modulo rational equivalence. I am looking for a reference for the following fact: If $X$ and $Y$ are smooth and projective varieties ove …