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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 …

I don't know what is going on exactly (misprints?), but here are some ideas: If you take a point of $q\in R_m$ (i.e. $U=Spec(k)$) defined by a sequence $0 \rightarrow G \rightarrow \mathcal{O}_X^{P(m) …

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 have two p.p. simple abelian surfaces $(A_i,D_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram: $\require{AMScd} \begin{CD} A_1 @>{birational}>> A_2\\ @V{2:1}VV @VV{2:1} …

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 …

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 …

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 we work over $\mathbb{C}$. Let $S\subset \mathbb{P}^3$ be a quartic surfaces with 16 nodes (ordinary double points). Then there is a simple principally polarized abelian surface $(A,\theta)$ …

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 am trying to understand the construction of Artin and Mumford of a non-rational unirational threefold in ([1], p.90). Assume $S$ is a smooth projective surface over $\mathbb{C}$ with a smooth curv …

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 …

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 …

Let $X$ be a smooth quasi projective variety over $\mathbb{C}$. Let $G$ be a finite abelian group acting via automorphisms on $X$. Denote by $G$-$\text{Hilb}(X)$ the subscheme of the Hilbert scheme o …

Assume $G$ is the Klein four group $G=\{1,\sigma_1,\sigma_2,\sigma_3\}$. Let $G$ act on $X=\mathbb{A}^2\times\mathbb{P}^1$ via: $$\sigma_1\cdot(x,y,[\lambda:\mu])=(-x,y,[\lambda:-\mu]) \text{ and } …

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 …