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 Dec 21 comment If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$ What if I take $X$ and $Y$ to be of dimension at least $2$? Then this phenomenon can't happen. Dec 21 asked If $X\times C$ is isomorphic to $Y\times C$, does it follow that $X$ is isomorphic to $Y$ Dec 20 accepted Abelian subvarieties of a principally polarized abelian variety are principally polarized Dec 20 revised Abelian subvarieties of a principally polarized abelian variety are principally polarized added 81 characters in body Dec 20 revised Abelian subvarieties of a principally polarized abelian variety are principally polarized added 366 characters in body Dec 20 asked Abelian subvarieties of a principally polarized abelian variety are principally polarized Dec 6 answered Shrinking the base field of a scheme of finite type over a field Dec 3 comment Are there infinitely many rational functions of bounded degree and given ramification @QiL No. I want to fix $X$. Then I want to find an integer $n$ and a finite set of points $R$ such that there are infinitely many $X\to \mathbf P^1(\mathbf C)$ of degree at most $n$ and with ramification inside $R$. For instance, if $g>1$, then $n=2$ won't work. (There is a unique hyperelliptic map $X\to \mathbf{P}^1$ if it exists at all.) You'll need to take $n$ "big enough" and $R$ also "big enough". (If I take $R$ small there aren't any $X\to \mathbf{P}^1(\mathbf{C})$ with ram locus inside $R$.) Dec 2 comment Are there infinitely many rational functions of bounded degree and given ramification I've edited the question. I admit the question was poorly written. Thanks for the comments. I hope it's more clear now. Dec 2 revised Are there infinitely many rational functions of bounded degree and given ramification Rewrote the question to make it more accessible Dec 2 comment Are there infinitely many rational functions of bounded degree and given ramification I just edited the question. I forgot to put the important condition that the rational function $X\to \mathbf{P}^1(\mathbf C)$ is ramified only inside $R$. Dec 2 revised Are there infinitely many rational functions of bounded degree and given ramification added 46 characters in body Dec 2 asked Are there infinitely many rational functions of bounded degree and given ramification Dec 2 comment Set of points where ring of germs is reduced is open This usually boils down to showing the following. Let $A$ be a local noetherian ring. Then, $A$ is reduced if and only if $A_p$ is reduced for all prime ideals $p$. Nov 30 comment Is a morphism of schemes which is proper at every fiber proper? Any open immersion $f:X\to Y$ of schemes which isn't closed will do, right? Nov 27 comment Are there infinitely many pairs of rational numbers such that… I like this answer a lot. It prompts the following question: Are there infinitely many square-free integers $n$ such that $nr^2=a^3+1$ has positive rank and $ns^2=b^3+2$ has positive rank? Nov 27 comment Closed morphism between schemes of finite type over a field induces a closed map between varieties? Thank you QiL! I only just read your comment. Nov 27 comment Are there infinitely many pairs of rational numbers such that… Question: Do I understand correctly that we always have $\mathbf{Q}(\sqrt{a^3+1}) =\mathbf{Q}(\sqrt{7}) = \mathbf{Q}(\sqrt{b^3+2})$ if $(a,r)$ and $(b,s)$ lie on the curve? Nov 26 answered Closed morphism between schemes of finite type over a field induces a closed map between varieties? Nov 25 revised When does a ring have a flat noetherian extension? deleted 45 characters in body