Mathematics Stack Exchange
Mathematics Stack Exchange

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
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Basic question: closure of irreducible sets

Let $(X,\tau)$ be a topological space, $U \subseteq X$ a non-empty open subset of $X$ and let $W$ be an irreducible non-empty closed subset of $X$. Assuming $W \cap U \neq \emptyset$ why is this intersection irreducible in $U$ and its closure (taken…
user6495
  • 3,957
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Cohomology of affine plane with double origin

How to calculate cohomology $H^1(X,O_X)$,$H^2(X,O_X)$ $H^1(X,O_X^*)$ of affine plane with double origin $X=\mathbb{A}^2\cup_{\mathbb{A}^2-\{0\}}\mathbb{A}^2$? To use Cech cohomology, I cannnot find a cover whose intersection all have trivial cover.…
user93417
8
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2 answers

Is a morphism between smooth varieties smooth if fibers are?

Suppose that $X$ and $Y$ are smooth varieties over a field $k$ (not necessarily algebraically closed), of dimension $m$ and $n$. Suppose we are given a morphism $\pi:X\rightarrow Y$. We know that if $\pi$ is smooth, it is flat (by definition) and…
Jakob Oesinghaus
  • 654
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Projective morphism defined by linear systems

Let $X$ be a normal variety and $D$ be a Cartier divisor, suppose $\sigma, \delta$ are two basepoint free linear systems in $|D|$, then we have two morphisms defined by these two linear system: $$\phi_{\sigma}: X \to \mathbb{P}^n,\qquad…
Li Yutong
  • 4,065
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4 answers

Determining the generators of $I(X)$

Let $ X \subset \mathbb A^3 $ be the union of three coordinate axes. How do I determine the generators of $I(X)$? Also, how do I show it has at least 3 elements?
Mohan
  • 14,856
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Finite surjective morphism of smooth varieties is flat

Let $f: X \to Y$ be a finite surjective morphism of nonsingular varieties over a field $k$. Exercise III 9.3. in Hartshorne's Algebraic Geometry sais that if $k$ is algebraically closed, then $f$ is flat. Is this still true, if $k$ is no longer…
Hans
  • 3,539
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Problem I.3.18 in Hartshorne

Problem I.3.18b-c in Hartshorne is concerned with the surface $Y$ of $\mathbb{P}^3$ given parametrically by $(x,y,z,w) = (t^4,t^3u,tu^3,u^4)$. In particular, part c asks to prove that $Y$ is isomorphic to $\mathbb{P}^1$. I did check two different…
Manos
  • 25,833
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1 answer

Picard group of a smooth projective curve

I have two (related) questions regarding the Picard group: 1) Are there examples of smooth projective curves with large Picard groups (say $Pic(X)\simeq\mathbb{Z}^n)$ for any $n$)? 2) In general, can we say what the Picard group is for a smooth…
adrido
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Problem I.3.13 in Hartshorne - Follow Up

Consider the context of this question: Local Ring of a Subvariety (problem 1.3.13 in Hartshorne). In trying to prove that $\dim \mathcal{O}_{Y,X} = \dim X + \dim Y$, I proved that $\mathcal{O}_{P,X} / m_{Y,X} \cong \mathcal{O}_{P,Y}$, where $P$ is…
Manos
  • 25,833
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votes
1 answer

Is $\mathbb{P}^2/C_n$ a variety?

Suppose we have a action of cyclic group of order $n$ on $\mathbb{P}^2$ by $k\to \{(x,y,z)\to(x,e^{2\pi ik/n}y,z)\}$. It has a fixed point and a fixed line. Is $\mathbb{P}^2/C_n$ a variety/scheme? If it is, can we write out its defining equations?…
user93417
7
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1 answer

How to make sense of the inverse image of a Weil divisor?

I was reading Hartshorne's proof that $Cl(X \times \mathbb A^n)=Cl(X)$. He defines the map $Cl(X) \to Cl(X \times A^n)$ by mapping the class of a prime divisor $D$ of $X$ to $p^{-1}(D)$ where $p:X \times A^n \to X$ the projection. I am not sure I…
Bernard
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4 answers

Show that a smooth plane quartic is never hyperelliptic

I have been asked to show that a smooth plane quartic is never hyperelliptic. I know that The genus of any such curve is 3 The statements of Riemann-Roch and Riemann-Hurwitz A curve is hyperelliptic if there's a degree 2 map from it to $\Bbb…
user3131035
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1 answer

When is a morphism of varieties which induces isomorphism on each fiber an isomorphism?

The question arises when I am considering something which is similar to the case of isomorphism of fiber bundles. Assume working over an algebraically closed field $k$. Let $E$ be a locally free sheaf of rank $N+1$ over Y, which is nonsingular, and…
Ziwen
  • 335
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A doubt in the proof of Prop. 1.10 of Hartshorne's Algebraic Geometry

I have a doubt in the proof of Proposition 1.10 of Hartshorne's book Algebraic Geometry, which states that if $Y$ is a quasi-affine variety, then its dimension is the dimension of its closure. In this proof the author picks a maximal chain $Z_0…
Diogenes
  • 643
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1 answer

An example of ample sheaf with no global section

In viewing the tags about ample bundle with no global sections I found an example below: If $C$ is a curve of genus $2$, and $p,q,r$ are general points on $C$, then the bundle $L=\mathcal{O}_C(p+q-r)$ is ample, but has no global sections at all.…
user93417
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