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One of the wholly unnecessary reasons that schemes are regarded with such fear by so many mathematicians in other fields is that three, largely orthogonal, generalizations are made simultaneously. C …

There are several rings-with-bases to get straight here. I'll explain that, then describe three serious connections (not just Ehresmann's Lesieur's proof as recounted in the OP). The wrong one is $Rep …

Toric varieties. They're so easy to define and work with, and to organize examples around. Like blowing up a scheme at a fat point, or blowing up in different orders, or big but not ample line bundles …

I very nearly wrote my PhD thesis on this topic. Here's as much as I was able to figure out, though it's hardly a direct answer to your question. 1) Say your total space is K\"ahler, and your fibers …

Let $A \to B$ be a map of commutative rings, and $d : B \to I/I^2$ be defined by $df = f\otimes 1 - 1\otimes f$, where $I$ is the kernel of $B \otimes_A B \to B$, as in [Hartshorne II.8]. If $df=0 …

Given a convex polytope whose facets are simplices, define the f-vector by f_i = the number of i-dim faces. Which vectors of integers are f-vectors? A list of conditions was conjectured, proven suffic …

Here's a place to see the normal cone side-by-side with other familiar constructions, that I learned from Fulton's "Intersection Theory". Here $X \subset Y$. Start with the space $Y \times {\mathbb P …

Bert Kostant mentioned an odd fact to me some time ago. As usual (with such statements), fix a complex, connected, reductive) Lie group $G$, with maximal torus $T$, and Weyl vector $\rho$ equal to ha …

Generic fiber vs. general fiber vs. geometric generic fiber.

I think you're absolutely right that the function $(i\in \mathbb N)\mapsto $interestingness($H^i$) is a rapidly decreasing function. I heard that Gel$'$fand compared it to the successive derivatives o …

Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not constructible!) function on $X$. I want to consider a Grot …

What these have in common is that they are of the form $G/P$ for $P$ a maximal parabolic. As such each has a minimal projective embedding of the form $G/P \hookrightarrow \mathbb P(V_\omega)$ where $V …

Let $R$ be a commutative ring, with whatever hypotheses let you answer the question -- e.g. Noetherian, local, finitely generated over $\mathbb C$. Let $I$ be the ideal defining the singular locus in …

In general, the only definition I know of GIT quotient is $Proj$ of the invariant ring. The obvious statements one can make about the rational map $Proj\ R\to Proj\ R^G$ are that it collapses $G$-orbi …

If you're willing to quotient by a nonreductive group, then $M_n//B$ will get you the $GL(n)$ flag manifold. (People are usually afraid to do so, worrying that the ring of invariants won't be Noetheri …