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Let us work over $\mathbb{C}$ for the moment. Assume we are given a real quadratic field $K$ with ring of integers $\mathcal{O}_K$. $\mathbf{Question:}$ Is there a smooth projective curve $C$ of gen …

Given a $\mathbb{C}$-scheme $S$, two $S$-schemes $X$ and $Y$ that are flat over $S$ and a coherent sheaf of $O_Y$-modules $F$. Assume we have a (faithfully) flat $S$-morphism $\pi: X \rightarrow Y$ a …

Assume we are given a simple abelian surface $A$ which has 2 non-equivalent principal polarizations $D_1$ and $D_2$ in $NS(A)$ (up to isomorphism), thus giving rise to two non-isomorphic smooth projec …

Assume $X$ is a smooth projective variety over $\mathbb{C}$ of dimension $n$, here $n\geq 3$, with a reduced normal crossing divisor $D\subset X$, such that $D=\sum\limits_{i=1}^r D_i$ where the $D_i$ …

Assume $\mathcal{A}$ is an Azumaya algebra of rank $r^2$ on a smooth projective scheme $Y$ over $\mathbb{C}$. Let $f: X\rightarrow Y$ be the Brauer-Severi variety associated to $\mathcal{A}$. I read …

Let $D\subset X$ be a smooth divisor on a smooth variety over $\mathbb{C}$. Then we have Gysin maps in étale cohomology $H^2(X\backslash D,\mu_2)\rightarrow H^1(D,\mathbb{Z}/2)$ as well as $H^2(k(X), …

Assume $S$ is a scheme over $\mathbb{C}$ (as nice as you want), $\mathcal{E}$ is a locally free $\mathcal{O}_S$-module and $\mathcal{A}$ is a coherent $\mathcal{O}_S$-algebra, locally free of finite r …

Assume $X$ is a noetherian scheme over $\mathbb{C}$ and $E$ a locally free sheaf of finite rank on $X$. Denote the the associated projective bundle by $f: \mathbb{P}(E)\rightarrow X$. Are there any c …

Your dimension is correct, this and more can be found in this article.

Assume we are given a nontrivial standard conic bundle $\pi: X\rightarrow S$, that is $X$ and $S$ are smooth projective varieties (say over $\mathbb{C}$), $\pi$ is flat and furthermore we have $Pic(X) …

Assume we work (over $\mathbb{C}$) on a polarized K3 surface $(X,L)$ with a line bundle $M$ on $X$ such that $M^2=-6$ and $ML=0$ as well as $h^0(M)=h^2(M)=0$ and thus $h^1(M)=1$. Then $E=\mathcal{O}_X …

Let $G=PGL(n)$ act on a smooth projective scheme $X$ over $\mathbb{C}$ with nontrivial finite stabilizers ($\cong \mathbb{Z}/2\mathbb{Z}$) only along a divisor $D\subset X$. Furthermore there a is a g …

Let $A$ be the $\mathbb{C}$-algebra generated by elements $i,j$ with relations $i^2=j^2=0$ and $ij=-ji$, i.e. we have $A=\mathbb{C}\oplus\mathbb{C}i\oplus\mathbb{C}j\oplus\mathbb{C}ij$. Let $\mathcal …

In Lemma 1.4. of this article, it is proven that $f$ flat, $X$ pure dimensional and $Y$ irreducible ensures that $X$ is reduced in your case. Maybe some of these requirments can be relaxed, I haven't …

Assume we are given a Brauer class $\xi\in Br(k(\mathbb{P}^n))$ ramified at some divisor $D\subset \mathbb{P}^n$, here $k=\mathbb{C}$. If $f: \mathbb{P}^n\mathrel{-\,}\rightarrow \mathbb{P}^n$ is a b …