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Let $K$ be a finite type field extension of $\mathbf{Q}$ and let $Y$ be a smooth quasi-projective variety over $K$. Let $\Omega_{Y/K}^{\bullet}$ denote the complex of sheaves of (algebraic) regular d …

Let $n\geq 2$ and $\mathbb{P}^n(\mathbf{C})$ be the complexe projective space of dimension $n$. Let $H\subseteq \mathbb{P}^n(\mathbf{C})$ be a hypersurface of degree $d$ where the coordinates in $\ma …

Let $X\subseteq \mathbb{P}^n(\mathbf{C})$ be a quasi-projective variety. Q: Is $X$ necessarily quasi-compact in the Zariski topology (if yes then how to prove it)?

So let $R$ be a discrete valuation ring and let $X$ be a scheme which is proper and flat over $R$. Let $X_s$ denote the special fiber of $X$. So intuitively, when somebody says that a curve $X$ is s …

Let $k$ be an arbitrary closed field (of arbitrary characteristic). Assume that we have a short exact sequence of k-algebraic abelian connected groups $$ 1\rightarrow K\rightarrow G \rightarrow H\rig …

Let $X\subseteq\mathbb{R}^k$ be a non-closed, unbounded semi-algebraic subset. Then it seems to me that Proposition 5.49 on p. 189 of the book Algorithms in Real Algebraic Geometry still holds true fo …

Let $A$ and $B$ be finitely generated $\mathbf{Z}$-algebra. Suppose that there exists two coprime integers $m$ and $n$ and an isomorphism of $\mathbf{Z}$-algebra $\phi:A\otimes_{\mathbf{Z}}\mathbf{Z}[ …

Let $p$ be a prime number and $X_0(p)/\mathbf{Q}$ be the classical modular curve for $\Gamma_0(p)$. Let $\tilde{X}_0(p)/\mathbf{Z}$ be the projective arithmetic surface corresponding to the normalizat …

Let $X$ and $Y$ be connected quasi-projective varieties over $\mathbf{C}$. Let $\mathcal{L}$ be an algebraic vector bundle over $X\times Y$. Let $p_2:X\times Y\rightarrow Y$ be the projection. ($\sta …

I have copied A. Stasinsky's comment who quoted a passage in the introduction of SGA1: "/.../ et de faire un ajustage terminologique, le mot morphisme simple ayant notamment \'et\'e remplac\'e entre- …

Let $X_0$ be a smooth projective variety over $\mathbb{C}$ and let $\Theta_{X_0}$ be the locally free sheaf of $O_{X_0}$-module corresponding to tangent space of $X_0$. Goal: To find a sufficient co …

Let $H_{\mathbf{Q}}$ and $H_{\mathbf{Q}}'$ be two pure Hodge structures of weight $n$ and $n'$ respectively. How do you prove the following simple fact: fact: If $n>n'$ and $f:H_{\mathbf{Q}}\rightarr …

What does it mean to say that a scheme $X$ is simple over $Spec(A)$ ? I stumbled on this terminology in a paper of S. Lubkin entitled "On a conjecture of Andre Weil".

Let $X$ be a smooth projective curve over a field $k$. We let $\omega$ be the canonical line bundle of $X$ and we denote by $F$ the field of $k$-valued rational functions on $X$. (1) When $k$ is alg …

Let $\pi:X\rightarrow C$ be a flat and proper morphism over $\mathbb{C}$ where $X$ is a smooth projective surface and $C$ is a smooth projective curve. Assume that all the fibers of $\pi$, except fini …