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Let $f:X \to Y$ be a flat morphism of projective noetherian integral schemes. Is there any known condition on a morphism $Z \to Y$ under which the resulting fiber product $X \times_Y Z$ is still integ …

Let $p:=[0,0,...,0,1] \in \mathbb{P}^n$ the point whose all the coordinates are zero except for the $n$-th. This defines a linear projection map $\phi:\mathbb{P}^n-p \to \mathbb{P}^{n-1}$, given by $[ …

Let $\phi:A \to B$ be a flat ring homomorphism, $M$ be a $B$-module which is flat when considered as an $A$-module. Is the tensor product $M \otimes_B M \otimes_B ... \otimes_B M$ flat over $A$? If no …

Let $f:X \to Y$ be a proper, flat morphism of noetherian schemes. Let $\mathcal{F}$ be a coherent sheaf on $X$ non-zero on every fibers of $f$. Is it true that $\mathcal{F}^{\vee \vee}$ is going to be …

Let $X$, $Y$ be (reduced) affine varieties and $f:X \to Y$ is a finite morphism which is an isomorphism over an open dense subset (for example a normalization map). Let $A$ be a local noetherian ring …

Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ …

Let $X$ be a (non-singular) complex surface and $(V,\nabla)$ be a vector bundle $V$ equipped with a flat connection $\nabla$ on $X$. Fix a point $x \in X$ and $v_0 \in V_x$ an element in the fiber ove …

Let $R$ be a discrete valuation ring and $X$ be a regular, integral. projective $R$-scheme, flat over $R$. Let $F$ be a torsion-free coherent sheaf on $X$ of rank $n$, flat over $\mathrm{Spec}(R)$. Is …

Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, projecti …