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Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms? Does a similar …

Suppose $X$ is a smooth projective surface with a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ over a field $k$, where all the fibres of $\pi$ are conics (i.e. a conic bundle). If $\pi$ admits …

Throughout, singular Del Pezzo means a surface with only isolated singularities and ample anti-canonical divisor. Suppose $X$ is a singular Del Pezzo of degree 2 over a field $k$ where $\text{char}(k) …

By a line I mean a (-1)-curve. Given a weak del Pezzo surface $X$ of degree $d$, how many lines would $X$ contain?

Let $X$ be a scheme and $\mathcal{F}$ a presheaf on $X_{ét}$. For each $x_{i}\in X$, pick a geometric point $\bar{x}_{i}$ over $x$ and denote by $i_{\bar{x}_{i}}:\text{Spec}(k_{i})_{\text{ét}}\rightar …

Let $X$ be a smooth cubic surface over a field $k$. Denote by $\bar{k}$ the separable closure of $k$ and $\bar{X}:=X\times_{k}\bar{k}$. Then it is well know that there exists a homomorphism $$ \phi:\t …

If a have a dominant morphism $\pi:X \rightarrow \mathbb{P}^{1}$ where $X$ is a projective, geometrically integral $k$-scheme. Then this gives rise to a pullback map \begin{align*} \pi^{*}:\text{Br}(k …

It is well known that when $K$ is a local or global field the Galois cohomology group $H^{3}(K,K_{\text{sep}}^{\times})=0$ where $K_{\text{sep}}$ denotes the separable closure of $K$. Could someone gi …

Given an extension $K = \mathbb{Q}(\alpha)$ by a cubic polynomial $g(x)\in \mathbb{Q}[x]$ (not necessarily Galois extension) is there a criterion for a cubic polynomial $f(x) \in K[x]$ to be reducible …