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Questions tagged [algebraic-geometry]

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

The study of geometric objects defined by polynomial equations, as well as their generalizations: algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc. Problems under this tag typically involve techniques of abstract algebra or complex-analytic methods. This tag should not be used for elementary problems which involve both algebra and geometry.

29074 questions
64
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Examples of morphisms of schemes to keep in mind?

What are interesting and important examples of morphisms of schemes (especially varieties) to keep in mind when trying to understand a new concept or looking for a counterexamples? Examples of what I'm looking for: The projection from the hyperbola…
user115940
  • 1,969
61
votes
2 answers

Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.

In chapter 2 section 7 (pg 151) of Hartshorne's algebraic geometry there is an example given that talks about automorphisms of $\mathbb{P}_k^n$. In that example Hartshorne states that $\mathcal{O}(-1)$ has no global sections. However, we know that…
MJoszef
  • 1,045
59
votes
5 answers

Geometric motivation for negative self-intersection

Consider the blow-up of the plane in one point. Let $E$ the exceptional divisor. We know that $(E,E)=-1$. What is the geometrical reason for which the auto-intersection of $E$ is $-1$? In general, what does it mean, geometrically, that a divisor has…
unk220
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41
votes
5 answers

Motivating Example for Algebraic Geometry/Scheme Theory

I am in the process of trying to learn algebraic geometry via schemes and am wondering if there are simple motivating examples of why you would want to consider these structures. I think my biggest issue is the following: I understand (and really…
Eric O. Korman
  • 19,137
36
votes
1 answer

What geometrical obstructions to $M$ being flat do elements which map to 0 in $M \otimes I$ represent?

I'm trying to get geometric intuition for the notion of a flat module over a ring, and am running into some problems with my intuition. I am comfortable with flat modules and tensor products from the commutative side, so when I ask what an object…
Hunter Spink
  • 1,699
32
votes
1 answer

Prove that sheaf hom is a sheaf.

Suppose $\mathcal{F}$ and $\mathcal{G}$ are sheaves on $X$. The sheaf hom from $\mathcal{F}$ to $\mathcal{G}$ is defined by $U \mapsto $ Hom($\mathcal{F}|_{U}$,$\mathcal{G}|_{U}$), where the Hom is taken in the category of presheaves, i.e.,…
user45955
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29
votes
1 answer

Equivalence of categories of vector bundles and locally free sheaves

Let $X$ be a scheme. It is known that the category $\mbox{Vec}_r(X)$ of vector bundles of rank $r$ on $X$ and the category $\mbox{Loc}_r(X)$ of locally free sheaves of rank $r$ on $X$ are equivalent (The equivalence is given by the functor $F\colon…
Irfan Kadikoylu
  • 1,016
28
votes
1 answer

How to think of the pullback operation of line bundles?

Recall that give a map $f : (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ of ringed spaces and a sheaf $\mathcal{F}$ on $Y$ we can form the pullback $f^\ast \mathcal{F} := f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}_Y} \mathcal{O}_X$. Now the pullback of…
user38268
27
votes
1 answer

Homotopy invariance of the Picard group

Is the Picard group of a scheme homotopy invariant in the sense that the projection $\pi : X \times \mathbb{A}^1 \to X$ induces an isomorphism $\mathrm{Pic}(X) \cong \mathrm{Pic}(X \times \mathbb{A}^1)$? Clearly it induces a split monomorphism, and…
Martin Brandenburg
  • 163,620
27
votes
3 answers

Why is the coordinate ring of a projective variety not determined by the isomorphism class of the variety?

I know that there are isomorphic projective varieties which have nonisomorphic coordinate rings, but I'm a little mystified as to "why" this is the case. Why doesn't a usual functoriality proof go through to prove this, and is there any insight into…
Tony
  • 6,718
26
votes
2 answers

How to compute the topological space of fibered product of schemes?

I know that the topological space of fibered product of schemes is generally distinct to the usual Cartesian product of toplogical spaces of schemes. Then how can we compute the top. sp. of fibered product of sch. explicitly? Is there any systematic…
User0829
  • 1,359
25
votes
2 answers

$\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample

Let $\mathcal{L},\mathcal{U}$ be invertible sheaves over a noetherian scheme $X$, where $X$ is of finite type over a noetherian ring $A$. If $\mathcal{L}$ is very ample, and $\mathcal{U}$ is generated by global sections, then $\mathcal{L} \otimes…
Li Zhan
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25
votes
1 answer

intuition on the projection formula

For a morphism $f:X\rightarrow Y$, locally free sheaf $\mathcal{G}$ on $Y$, and a quasi-coherent sheaf $\mathcal{G}$ on $X$, we have the projection formula $$f_*(\mathcal{F}\otimes_{\mathcal{O}_X}f^*\mathcal{G})\simeq…
adrido
  • 2,283
25
votes
2 answers

Simple example of an ample line bundle that is not very ample

I am looking for a very concrete and simple example of a line bundle $L$ (on a curve or a surface) which is ample, but not very ample. I would also like that $L^{\otimes k}$ is very ample for a small $k$, in the sense that I can do a very hands-on…
Jesko Hüttenhain
  • 14,792
24
votes
2 answers

What are some applications outside of mathematics for algebraic geometry?

Are there any results from algebraic geometry that have led to an interesting "real world" application?
Jonathan Fischoff
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