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Let $X$ be a projective variety over $\mathbb C$ of dimension $n$. Let $\tilde{X} \to X$ be a resolution of singularities. Suppose that $H^n(\tilde{X}, \mathcal O_{\tilde{X}}) = 0$. What can we say …

A uniruled variety is the one which admits a dominant map from $X \times \mathbb P^1$. I think it is true that uniruled varieties are rationally connected. Is the converse true? What about low dime …

In an answer to a MathOverflow question on the following link Vector bundles on $\mathbb{P}^1\times\mathbb{P}^1$, it is mentioned that $\mathbb P^1 \times \mathbb P^1$ has an Ulrich sheaf. However, …

Let $A$ be a seminormal ring. (Assume that $A$ is a finitely generated $k$-algebra, if it helps.) Is it true that $A$ is regular in codimension 1? I know this is true for normal rings. If the answ …

We will always work with finite-type, smooth schemes over a field $k$. Let $\pi: Y \to X$ be an etale map of $k$-schemes. Let $Z$ be another $k$-scheme admitting a morphism $f: Y \to Z$. Suppose th …

Let us work in the category of smooth, projective varieties (say, over an algebraically closed field $k$). If $X$ and $X'$ are birational, then do they have the same Chow groups? Is there at least a …

I know that not every local seminormal ring is Cohen-Macaulay. But are 1-dimensional local seminormal rings Cohen-Macaulay?