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$\DeclareMathOperator{\Pic}{Pic}$Let $C$ be an integral projective curve over $\mathbb C$, which is smooth except for a single node $x\in C$. Let $M$ be the moduli space of stable torsion-free sheaves …

$\DeclareMathOperator{\Tors}{Tors}\DeclareMathOperator{\Pic}{Pic}$Here is what I got from discussing with my advisor. Basic construction Let $L$ be a line bundle on $\tilde C$, and let $\pi: (\nu_* L) …

The following came up when reading the definition of the moduli stack of principally polarized abelian varieties in [1]. Let $\pi_1:A_1 \to S_1$ and $\pi_2: A_2 \to S_2$ be abelian schemes over $S_i$, …

$\DeclareMathOperator{\Jac}{Jac}\DeclareMathOperator{\Pic}{Pic}$Let $C$ be a smooth curve of genus $g > 0$, and consider the Picard torus $\Pic^d(C)$ of line bundles of degree $d$. Let $\mathcal P$ be …

$\DeclareMathOperator{\Pic}{Pic}\DeclareMathOperator{\Sym}{Sym}$I found an answer in a paper by Schwarzenberger[1]. Fix a point $c \in C$, and for $n \in \mathbb Z$ consider on $\Pic^0(C)$ the sheaves …

Take a point $x \in X$ and set $y = f(x) \in Y$. Let $X_y = f^{-1}(y)$ the "scheme-theoretic" fiber, i.e. the complex-analytic subspace of $X$ which is cut out the equation $f(x) = y$. Your question …

Let $X \subset Y$ be an inclusion of compact complex (possibly Kähler) manifolds. I'm wondering if "Riemann-Roch without Denominators" [1, Thm 15.3] holds in that situation. The statement is For a ve …

In my research I encountered automorphisms of the ring of convergent power series $$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$ which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm wonderi …

I have a proper map of complex manifolds $$f: X \to D,$$ where $D \subset \mathbb C$ is the unit disc. By assumption, $f$ has connected fibers, is smooth over $D \setminus 0$, and a smooth fiber $F$ i …

Oguiso writes[1] Theorem 1.1 Let $f: X \to \mathbf P^n$ be an abelian fibered HK [hyperkähler] manifold. Let $K = \mathbf C(\mathbf P^n)$ and let $A_k$ be the generic fiber of $f$. Then, $\rho(A_K)= …

I already asked this on Math.SE, but didn't receive an answer yet. Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an effecti …

I managed to calculate my examples. In the first one, $D_0$ has indeed polarization type $(2,2)$. To see this, let $E = \mathbb C / (\mathbb Z + \tau \mathbb Z)$ be an elliptic curve, and consider the …

Let $S \to \mathbb P^2$ be a two-to-one cover branched over a sextic, i.e. $S$ is a K3-surface. Let $C \subset S$ be the preimage of a (smooth) quadric, so that by Hurwitz' formula, $g(C) = 5$. Accord …

I already asked this on math.stackexchange.com, but didn't receive an answer. Suppose $f: X \to Y$ is a (possibly proper) morphism of complex manifolds (resp. smooth varieties) such that all fibers of …

I already asked this on math.SE, but didn't receive any response. The following question arose when studying Hwang and Oguiso's Characteristic foliation on the discriminant hypersurface of a holomorph …