The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1).It was finalized by Alexandre GROTHENDIECK, during the 1950's and ...

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11
votes
1answer
1k views

What are the monomorphisms in the category of schemes?

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...
11
votes
4answers
2k views

Extending vector bundles on a given open subscheme

Many people seem to know the following. Personally, I don't quite understand it though and maybe I'm wrong. It's the fact that "a vector bundle on an open subscheme extends in only one way to a vector ...
13
votes
3answers
837 views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
9
votes
2answers
349 views

Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
3
votes
1answer
247 views

Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
3
votes
0answers
225 views

Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces. Under these ...
2
votes
0answers
73 views

Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature. To be more clear let me explain the example I have in mind. Let ...
0
votes
1answer
93 views

curve through a point avoiding an hypersurface

Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset. Let $x\in H$, Is it possible to find an curve ...