Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.
Questions tagged [ag.algebraic-geometry]
21563 questions
141
votes
0 answers
Grothendieck -sad news
Sorry for that this is not a real question. But I thought people would like to know.
Alexandre Grothendieck died today: http://www.liberation.fr/sciences/2014/11/13/alexandre-grothendieck-ou-la-mort-d-un-genie-qui-voulait-se-faire-oublier_1142614
DamienC
- 8,113
127
votes
15 answers
A learning roadmap for algebraic geometry
Unfortunately this question is relatively general, and also has a lot of sub-questions and branches associated with it; however, I suspect that other students wonder about it and thus hope it may be useful for other people too. I'm interested in…
Akhil Mathew
- 25,291
87
votes
15 answers
The importance of EGA and SGA for "students of today"
That fact that EGA and SGA have played mayor roles is uncontroversial. But they contain many volumes/chapters and going through them would take a lot of time, especially if you do not speak French.
This raises the question if a student, like me,…
user1161
- 193
82
votes
5 answers
how does one understand GRR? (Grothendieck Riemann Roch)
I tried to answer an earlier question as to uses of GRR, just from my reading, although i do not understand GRR. Today i tried to understand the possible idea behind GRR. After editing my answer accordingly, it occurred to me i was asking a…
roy smith
- 12,063
68
votes
12 answers
Life after Hartshorne (the book, not the person)...
I was wondering what material in algebraic geometry is crucial and is a logical step for a serious graduate student in algebraic geometry once they've finished Hartshorne. Good answers could include a list of areas of algebraic geometry or…
anonymous
- 191
55
votes
1 answer
Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?
Possibly this has already been asked, but it came up again in this question of Daniel Litt. Does every smooth, projective morphism $f:Y\to \mathbb{C}P^1$ admit a section, i.e., a morphism $s:\mathbb{C}P^1\to Y$ such that $f\circ s$ equals…
Jason Starr
- 4,091
53
votes
6 answers
Colimits of schemes
This is related to another question.
I've found many remarks that the category of schemes is not cocomplete. The category of locally ringed spaces is cocomplete, and in some special cases this turns out to be the colimit of schemes, but in other…
Martin Brandenburg
- 61,443
49
votes
6 answers
Open affine subscheme of affine scheme which is not principal
I'm not sure whether this is non-trivial or not, but do there exist simple examples of an affine scheme $X$ having an open affine subscheme $U$ which is not principal in $X$? By a principal open of $X = \mathrm{Spec} \ A$, I mean anything of the…
Wanderer
- 5,123
47
votes
1 answer
Did Grothendieck have a plan for proving Riemann Existence algebraically?
A recent question reminded me of a question I've had in the back of my mind for a long time. It is said that Grothendieck wanted the center-piece of SGA1 to be a completely algebraic proof (without topology) of the following…
James D. Taylor
- 6,178
45
votes
3 answers
Intuition behind Riemann's bilinear relations
I'm trying to understand Riemann's bilinear relations on the normalized period matrix of a Riemann surface. Recall that they say the following. Let $X$ be a compact Riemann surface of genus $g>0$. Fix a standard basis $\{a_1,b_1,\ldots,a_g,b_g\}$…
BFA
- 451
44
votes
1 answer
Possible formal smoothness mistake in EGA
EGA IV 17.1.6(i) states that formal smoothness is a source-local property. In other words, a map $X\to Y$ of schemes is formally smooth if and only if there is an open cover $U_i$ ($i\in I$) of $X$ such that each restriction $U_i\to Y$ is formally…
JBorger
- 9,278
42
votes
3 answers
What does the Lefschetz principle (in algebraic geometry) mean exactly?
This principle claims that every true statement about a variety over the complex number field $\mathbb{C}$ is true for a variety over any algebraic closed field of characteristic 0.
But what is it mean? Is there some "statement" not allowed in this…
stjc
- 1,072
40
votes
9 answers
Why must nilpotent elements be allowed in modern algebraic geometry?
On the Wikipedia page1 about algebraic varieties https://en.wikipedia.org/wiki/Algebraic_variety, a sentence reads as follows:
[[A more significant modification is to allow nilpotents in the sheaf of rings. A nilpotent in a field must be 0: these if…
minimax
- 1,137
39
votes
1 answer
$V$, $W$ are varieties. Does $V\times \mathbf{P}^1=W\times \mathbf{P}^1$ imply $V=W$?
If $\mathbf{P}^1$ is replaced by the affine line $\mathbf{A}^1$, this becomes the cancellation problem, and we have a pair of famous Danielewski surfaces ($xy=1-z^2$ and $x^2y=1-z^2$) as a counterexample (though I'm still seeking how to prove…
Honglu
- 1,079
39
votes
7 answers
Geometric meaning of the Euler sequence on $\mathbb{P}^n$ (Example 8.20.1 in Ch II of Hartshorne)
Is there any geometric way to understand the exact sequence in Example 8.20.1 in Ch II of Hartshorne (p. 182), or its dual from theorem 8.13?
Here is the sequence:
$0\to O_{\mathbb{P}^n}\to O_{\mathbb{P}^n}(1)^{n+1}\to T_{\mathbb{P}^n}\to 0$
Enrique Acosta
- 1,209