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I was this past year working with a bright high-schooler on algebraic geometry following Reid's book Undergraduate Algebraic Geometry, and we got all the way to proving that there is at least one line ...

I learned some basic properties of Hirzebruch surface mainly from Vakil's notes "the rising sea", section 20.2.9. the Hirzebruch surface is defined as $\mathbb{F}_n:=\operatorname{Proj} (\...

For a compact Riemann surface $B$ of genus $\geq 2$, it is a consequence of the Narasimhan-Seshadri theorem that there exist rank-$2$ vector bundles $E \to B$ of degree zero, all of whose symmetric ...

Let k be an algebraically closed field. It's well known that every complete curve, period, is projective. Also, that every smooth surface is, and that there are smooth 3-folds which are not, and ...