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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
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Powers of a sheaf of ideals

Let $\mathscr I$ be a sheaf of ideals on a scheme $X$. What is the meaning of $\mathscr I^2$? I would think we would define $\mathscr I^2(U)$ to be the ideal $\mathscr I(U)\mathscr I(U)$. But there is no reason to believe this defines a sheaf. Is…
D_S
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Vakil's Foundations of Algebraic Geometry, Exercise 7.3.F

$\DeclareMathOperator{\Spec}{Spec}$ I'm having trouble with the exercise in the title, which states "Suppose $Z$ is a closed subset of an affine scheme $\Spec A $ locally cut out by one equation. (In other words, $\Spec A$ can be covered by smaller…
Diego Fioravanti
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Are sets given in parametric form always algebraic?

If a set is given in parametric form by polynomials, is this set always closed (Zariski topology), i.e algebraic? For example, take $X=\{(t,t^{2},t^{3}): t \in \mathbb{A}^{1}\}$ and $W=\{(t^{3},t^{4},t^{5}):t \in \mathbb{A}^{1}\}$ one can check…
user10
  • 5,688
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Must a proper curve minus a point be affine?

Let $C$ be a proper smooth geometrically connected curve over a field $K$, and let $P\in C(K)$ be a point. Must $C - P$ be affine? EDIT: By Riemann-Roch, you can definitely find functions $f_1,\ldots,f_r : C-P\longrightarrow\mathbb{A}^n_K$, but how…
oxeimon
  • 12,279
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Why morphism between curves is finite?

If $X$ is a complete nonsingular curve over $k$, $Y$ is any curve over $k$, $f: X \to Y$ is a morphism not map to a point (so $f(X)=Y$), then $f$ is a finite morphism. This is the assertion prove in Hartshorne Chapter2, Prop6.8. But the proof is a…
Li Zhan
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cusp and node are not isomorphic

Let $k$ be an algebraically closed field of characteristic $\neq 2$ and consider the affine varieties $X_1 := V(x^2-y^3)$ (the cusp) and $X_2 = V(y^2-x^2 (x+1))$ (the node). I would like to show that $X_1$ and $X_2$ are not isomorphic. This is easy…
Martin Brandenburg
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Relations between the fiber of a vector bundle and the stalk of the corresponding sheaf

I know that the fiber of a vector bundle and the stalk of the corresponding sheaf are different objects. Shafarevich says that there's a general relation between this fiber and the aforementioned stalk, that's to say…
TheWanderer
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Curve in $\mathbb{A}^3$ that cannot be defined by 2 equations

According to a problem in Shafarevich 1.6, every curve in $\mathbb{A}^3$ can be cut out by 3 equations. Can someone give me an example of a curve in $\mathbb{A}^3$ that is not cut out by 2 equations?
Tony
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Surjective morphism of affine varieties and dimension

I'm trying to do the following exercise from the book An Invitation to Algebraic Geometry: Show that if $X \to Y$ is a surjective morphism of affine algebraic varieties, then the dimension of $X$ is at least as large as the dimension of $Y$. Could…
Math536
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Are the concepts of scheme-theoretic complete intersection and Ideal theoretic complete intersection the same?

Suppose $X$ is a projective variety in $\Bbb{P}^n$ of dimension $k$. Everything is over a field. We say $X$ is a scheme-theoretic complete intersection if $X$ can be written as $V_+(f_1) \cap \ldots \cap V_+(f_{n-k})$ hypersurfaces where the…
user38268
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2 answers

Jacobian criterion for projective varieties

Let $P\in Y=Z(f_{1},\cdots ,f_{s})$ be a projective variety. Then $Y$ is non singular at $P$ if and only if rank of the matrix $\Vert\partial f_{i}/x_{j}(P)\Vert=n-\dim{Y}$. I know of the statement for affine varieties, and I am trying to prove it…
TheNumber23
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Is a morphism of reduced schemes over an algebraically closed field determined by its values on closed points?

Let $X$ and $Y$ be reduced schemes over an algebraically closed field $k$ of positive characteristic. Suppose $f$ and $g$ are morphisms $X \to Y$ with $f(x) = g(x)$ for every $k$-point $x$ of $X$. Does $f = g$ as morphisms of schemes?
Dilbert H.
  • 113
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What fails in the Cartier <-> Weil divisor correspondence in the singular case?

I know that the following holds in much more generality, but lets say everything happens in the toric case over $\mathbb{C}$. Setting: Given a smooth variety $X$ then there is an isomorphism between the group of Cartier divisors of $X$ and the group…
Johannes
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Why did Hartshorne bother on schemes in his textbook when he worked only over an algebraically closed field?

In his textbook Algebraic Geometry, he wrote in p. 58: Now that we have seen a litle bit of what algebraic geometry is about, we should discuss the degree of generality in which to develop the foundations of the subject. In this chapter we have…
Makoto Kato
  • 42,602
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Is the set of closed points of a $k$-scheme of finite type dense?

Let $k$ be a field. Let $X$ be a scheme of finite type over $k$. We denote by $X_0$ the set of closed points of $X$. Is $X_0$ dense in $X$? Motivation See my comment to Martin Brandenburg's answer to this question.
Makoto Kato
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