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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 $D_i$ …

I cannot comment due to low reputation, but there is a whole chapter on this in "Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn. There are also some explicit computations to compare the …

Let $X$ and $T$ be schemes and assume we have two coherent sheaves $\mathcal{F}$ and $\mathcal{G}$ on $X\times T$ which are flat over $T$, that is these are families of sheaves parametrized by $T$. …

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 $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$. Assume the canonical morphism $F\xrightarrow{\sim} f_{*}f^{*}F$ is …

Assume we have two standard conic bundles $\pi:C \rightarrow X$ and $\pi': C'\rightarrow X'$. That is $\pi$ and $\pi'$ are flat morphims of smooth varieties over $\mathbb{C}$ and both maps are relativ …

I cannot comment due to reputation and haven't done the computations, but I see two possible problems: usually one has a spectral sequence $(R^if_{*})(\mathcal{E}xt^j(M,N)) => \mathcal{E}xt_f^{i+j}( …

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 …

Assume we have the projective plane $\mathbb{A}^2=Spec(\mathbb{C}[r,s])$. Now take the projective plane over this affine plane $\mathbb{P}^2_{\mathbb{A}^2}$ with homogenous coordinates $[u:v:w]$. Def …

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 …

Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$. Now $f$ is a flat mo …

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 …

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 …