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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.

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Showing that $X \times \mathbb{A}^1$ is regular in codimension one.

I am reading the proof of Proposition 6.6 in Hartshorne which states that $\operatorname{Cl} X \cong \operatorname{Cl} (X \times \mathbb{A}^1)$. Let $X$ be a noetherian, integral, separated scheme which is regular in codimension one. We want to…
user7090
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What morphisms of schemes fulfill this property?

Let $f:Y\to X$ and $g:Z\to Y$ be morphisms of schemes and take the product $Z\times_X Y$ with respect to the composition $g\circ f:Z\to X$ and $f$. What property of a scheme morphism does $f$ have to fulfill such that the projection $Z\times_X Y\to…
Daniel Dreiberg
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Plane intersection by a mapping and different cases

In $\mathbb{R}^3$ say we have the 2 planes $A=\{z=1\}$ and $B=\{x=1\}$. A line through 0 meeting $A$ at $(x,y,1)$ meets $B$ at $(1,y/x,1/x).$ Consider the map $\phi: A \rightarrow B$ defined by $(x,y) \mapsto (y' = y/x, z' = 1/x)$. I'm trying to…
mary
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Open subscheme of an irreducible component remains open?

Let $X$ be a scheme (e.g. of finite type over $\mathbb C$, but it does not matter) and let $Z\subset X$ be an irreducible component of $X$. Suppose we have an open subscheme $U\subset Z$. How to characterize when $U$ will still be open in $X$? The…
Brenin
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The affine coordinate ring of twisted cubic curve, $Y$ is $A(Y)=k[x,y,z]/(z-x^3, y-x^2)$?

I am working on the following problem: Let $Y \subset A^3$ be the set $Y={(t,t^2,t^3)|t\in k}$ ($A^3$ is the affine 3-space over $k$ an algebraically closed field.) Show that $Y$ is an affine variety of dimension 1. Find generators for the ideal…
Mathemphetamine
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What's a morphism? Well it's a morphism.

I'm confused on the definition of an "$F$-morphism" of $F$-varieties. The textbook is Springer, Linear Algebraic Groups. Let $k$ be an algebraically closed field, and $F$ a subfield of $k$. The notions of an affine $F$-variety, and of a morphism…
D_S
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Extending morphisms between varieties

This is Exercise 5.8 from Gathmann's notes on Algebraic Geometry, and I'm having a bit of trouble for (a) and (b): page 41 of http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2014/main.pdf (a) asks you to show that any morphism…
whetham
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Is it enough to check corank of jacobian matrix at closed points

This is actually exercise 12.2.H of Vakil's notes. In the notes, a k-scheme is defined to be k-smooth of dimension d if there exists a affine open cover(every is of form $A=k[x_1,...,x_n]/(f_1,...,f_r)$) where the Jacobian matrix has corank d at all…
user330928
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Need a counterexample to show that Cl$(X\times Y)$ is not always same as Cl$(X) \oplus $Cl$(Y)$

Recall that for a quasi projective variety $X$ one can define the Divisor Class Group denoted by $\operatorname{Cl}(X)$. Suppose $X$ and $Y$ be two quasi projective varieties.What is the counterexample to show that $\operatorname{Cl}(X\times Y)$ is…
Arpit Kansal
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Are closed points of a scheme $\frac{X}{k}$ the same $\overline{k}$-points, modulo Galois group action

Let $k$ be a field, and $X$ a scheme locally of finite type over $k$. Let $\overline{k}$ be the algebraic closure of $k$. Is it true that the set of closed points of $X$ is in bijection with $$\frac{X(\overline{k})}{Gal(\frac{\overline{k}}{…
usr0192
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Some clarifications regarding the definition of the Hilbert-Mumford weight

I am reading about the Hilbert-Mumford criterion and I am stuck at something that is stated as "obvious" in every text that I can find. A bit of help would be much appreciated. So, let $X$ be a projective variety over a field $k$, $G$ a reductive…
baltazar
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Hypersurfaces have no embedded points (Vakil 5.5.I)

Here's a question from Vakil's FOAG. If $f\in k[x_1,\ldots,x_n]$ is non-zero, show that $A:=k[x_1,\ldots,x_n]/(f)$ has no embedded points. Hint: suppose $\bar{g}\in A$ is a zero-divisor, and choose a lift $g\in k[x_1,\ldots,x_n]$ of $\bar{g}$. Show…
Houndoom
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Hartshorne 4.1.6 Gonality of a curve

I have a question about the following exercise from Hartshorne's book 'Algebraic geometry': Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f:X\rightarrow \mathbb P^1$ with degree $\leq g+1$. My idea is the following: We…
slin0
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Hartshorne Lemma V.1.3 meaning of exact sequence

I've been trying to make sense of the exact sequence in Lemma 1.3 chapter 5. The Lemma is the following: Let $C$ be a smooth irreducible curve on a smooth projective surface X, and let $D$ be any curve meeting $C$ transversally. Then $$\# (C \cap…
Daniel Levine
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Abundence of smooth curves on a normal variety?

If $X$ is a normal variety, and $p \in X$, is it true that there is a curve $C \subset X$ with $p \in C$ a smooth point (on C)? It is obviously false if normality is dropped - take $X$ to be a singular curve. I have no specific reason beyond this to…
Elle Najt
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