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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Does a motive capture everything about an algebraic variety?

Is the functor from the category of projective varieties over a field $k$ to the category of pure motives over $k$, faithful? (Perhaps it is not full). Ditto: Is the functor from the category of affine varieties over a field $k$ to the category of…
William
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1 answer

Degree of a morphism of projective curves

I think this is going to be a silly question. I'm happy with the following fact: If $\alpha : X \to Y$ is a non-constant morphism of irreducible curves, then $\alpha$ induces an embedding of field $k(Y) \hookrightarrow k(X)$ such that $[ k(X) : k(Y)…
Jonathan
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6
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1 answer

degree of an etale cover of the affine line

Let $X\subset \mathbb{A}^N_k$ be an irreducible smooth variety over an algebraically closed field $k$. Suppose we have an etale map $\pi:X\to \mathbb{A}^1_k$. Are there any bounds on the degree of $\pi$? Here etale means flat with smooth, finite…
adrido
  • 2,283
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1 answer

Hypersurface becomes an hyperplane after embedding

Let $X$ be an hypersurface of degree $k$ in $\mathbb{P}^{n}$, why the equation defining $X$ becomes linear in the Veronese coordinates? More precisely I want to understand the last paragraph of the following…
user10
  • 5,688
6
votes
1 answer

composing with a non-flat morphism

Let $X,Y,Z$ be integral schemes and $f:X\to Y$, $g:Y\to Z$ be morphisms such that $g$ is flat and $f$ is not flat. Does this imply that the composition $g\circ f:X\to Z$ is necessarily non-flat? [EDIT: ykm's example works but I'm also interested to…
adrido
  • 2,283
6
votes
4 answers

Complement of a point in $\mathbb{P}^{2}$

This is question $5$ from Shafarevich's book page $66$. Let $X=\mathbb{P}^{2} \setminus x$ where $x$ is a point. Prove that $X$ is not isomorphic to affine nor a projective variety. How to prove this?
user6495
  • 3,957
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Exercise on Fulton's "Algebraic Curves"

Exercise 7.12 from Fulton's Algebraic Curves Find a quadratic transformation of $\; F = Y^2 Z^2 − X^4 −Y^4$ with only ordinary multiple points. By checking the partial derivatives, I found that $P=[0,0,1]$ is a singular point. Checking the…
AnalysisStudent0414
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j-invariants for an elliptic curve over the Artin ring $k[t]/(t^n)$.

Let $k$ be an algebraically closed field, let $\lambda \in k - \{0,1\}$ and let $C = k[t]/(t^n)$. Hartshorne's "Deformation Theory" chapter 1 exercise 4.9(c) asserts that the family $$y^2 = x(x-1)(x-(\lambda +t))$$ is a nontrivial family over $C$.…
beeflavor
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2 answers

A curve has infinitely many points

Let $f\in k[x,y]$, where $k$ is an algebraically closed field. I would like to prove the curve $f(x,y)=0$ has infinitely many points. What I know is $k$ is infinite, but I don't know how to use this to prove this curve has infinite…
user42912
  • 23,582
6
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1 answer

Connected vs irreducible Variety

I am asking if there is any particular criterion for a connected component of given variety to be irreducible (you can assume suitable conditions on the variety) thanks
Z.A.Z.Z
  • 547
6
votes
1 answer

natural structure morphism from Proj $S$ to Spec $A$

Let $S$ be a finitely generated graded $A$ algebra, where $A$ is a commutative ring with unity. The exercise says to describe a natural structure morphism from Proj $S$ to Spec $A$. I would appreciate some assistance! Thanks!
user211392
  • 767
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1 answer

Divisor on surface

I'm trying to understand the following result: Let $S$ be a smooth, projective surface over $\mathbb{C}$ and let $D$ be a divisor on $S$. Let $H$ be a hyperplane section of $S$ for a given embedding. Then for some $n \geq 0$, we can write $D \equiv…
Evariste
  • 2,620
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1 answer

True or False: $f$ is injective if and only if $f^*$ is surjective where $f^*$ is corresponding to the pullback to $f$.

Let $f: X\rightarrow Y$ be a morphism of affine varieties and $f^*: A(Y)\rightarrow A(X)$ the corresponding homomorphism of the coordinate rings. The question is whether this is true or false: $f$ is injective if and only if $f^*$ is surjective.…
KittyL
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6
votes
1 answer

Isomorphic elliptic curves are projectively equivalent

Let $E_1$, $E_2 \subset \mathbb{P}^d$ be two smooth elliptic curves, that are isomorphic as abstract curves. How can one prove that they are projectively equivalent? That is there is a automorphism $\phi \in \mathbb{P}GL(d+1)$ s.t. $\phi(E_1)=E_2$.
Alex
  • 6,264
6
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0 answers

Blowup of $\mathbb{P}^1\times\mathbb{P}^1$ and a sheaf computation

Let $Y_1=Y_2=\mathbb{P}^1$, $Y=Y_1\times Y_2$, $p_i:Y\rightarrow Y_i$, $i=1,2$ be a canonnical projections and $\pi:X\rightarrow Y$ be a blowup of $Y$ in a finite set of points. How to compute $(p_2\pi)_*(p_1\pi)^*(\mathcal{O}_{Y_1}(1))$? I tried to…
cmplx_variable
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