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I think there could be different definitions of what a divisor is, but if want to keep the property that divisors are connected to line bundles, you are forced to relate to combinations of subvarietie …

How do you define blow-up? It should be straightforward to show that an explicit construction has relative dimension 0 over $X$. Update: In the comments I suggest to take the formal two-dimensional …

is there a way to glue two schemes together along a closed point (say we're working over a field)? Is it easier to glue in the category of algebraic spaces? For this particular pushout, the geome …

Indeed, as follows from the comments below, maps between schemes provide examples of Fourier-Mukai transform, most famous example being a similar map with additional twisting by a bundle in $A\times \ …

I had this simple question when formulating the Todd class question. Does there exist an example of proper morphism $f:X\to Y$ together with nontrivial homology class $t\in H^*(X)$ such that for a …

If you browse MO pages, you'll notice there are many references to SGA and EGA, recommended by professionals :) Though reading it fully in itself might be not the best idea — there are lots of great …

Preliminaries First part of your question doesn't use the bialgebra structure. That is, you have a space of functions on countable many points which I'll denote $A = \mathbb C_1\times \mathbb C_2\tim …

In standard topological terms, the exact sequence that relates homotopy groups of the base $B$, fiber $F$ and total space $E$ of topological fibration gives $$\pi_1(F) \to \pi_1(E) \to \pi_1(B) \to \ …

I might be missing something about question 3. Here's a simple construction: Consider a projective space $P$ of dimension $\text{dim}\\, P_1 + \text{dim}\\, P_2$ that contains both $P_1$ and $P_2$ in …

It's an interesting question, but it's not obvious knowing a function will help you a lot. E.g. consider a similar question: it feels bad talking about elliptic curve without explicitly writing its eq …

It may be true that most books start with commutative algebra, but I think that it's possible to teach alggeom with simple pedagogical examples. My favorite one is about the definition of a point for …

The definition in Hartshorne is, surprisingly, the most natural one. Let's approach this in a following way. If you meditate about what Grothendickian algebraic geometry is, you grok suddenly that t …

First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction. This was actually forward-referring to C …

My memory is fuzzy, but when you compute algebraic bundle cohomology, and it happens to be 0 here because of GAGA, I think, on all affine subschemes, wouldn't it automatically mean the bundle is local …

Here's a typical example of $M$ with the property that $O_x = M_x$ for $x$ in open subset. Take $U = \mathop{\mathrm{Spec}}A-\{f=0\}$. Note that $U$ is $\mathop{\mathrm{Spec}} A_f$ where $A_f$ is a l …