For AlgGeom questions related to varieties, with or without the methods of scheme theory for their formulation or solution. Examples of the 'without' case are variety theory as understood by Serre in FAC and GAGA, or the study of algebraic subsets of the classical affine and projective spaces ₖⁿ=kⁿ, ℙₖⁿ=ₖⁿ⁺¹/k*, as exemplified in books such as W. Fulton, Algebraic Curves, J. Harris, Algebraic Geometry: a first course, or J. S. Milne, Algebraic Geometry
Questions tagged [classical-algebraic-geometry]
17 questions
3
votes
1 answer
Is $\mathbb{C}^{n+1}$ \ $\{0\}$ isomorphic to $\mathbb{P}^n\times \mathbb{C}^*$?
Is $\mathbb{C}^{n+1}$ \ $\{0\}$ isomorphic to $\mathbb{P}^n\times \mathbb{C}^*$?
Where $\mathbb{P}^n$ is $n$-dimensional complex projective space, $\mathbb{C}^*=\mathbb{C}$ \ $\{0\}$.
My idea:
Under the classical topology, the…
clgdj
- 303
2
votes
1 answer
Do two lines passing through two (not necessarily distinct) points on an irred. cubic necessarily coincide? (Proof of Sec. 5.6, Prop. 3 in Fulton)
Prop. 3 of Sec. 5.6 of Fulton's "Algebraic Curves" is:
Let $C$ be an irreducible cubic, $C'$, $C''$ cubics. Suppose $C' ⋅ C = ∑_{i=1}^9 P_i$, where the $P_i$ are simple (not necessarily distinct) points on $C$, and suppose $C'' ⋅ C = ∑_{i=1}^8 P_i…
Fred Akalin
- 869
2
votes
2 answers
How do I translate the intersection of two affine curves in a plane into a statement about ideals in $k[X, Y]$?
Let $E : y^2 = x^3 + Ax + B$ be an elliptic curve over a field $k$ with characteristic not $2$ or $3$ (not necessarily algebraically closed), and let $x = x_0$ be a line that intersects $E$ in $(x_0, \pm y_0) \in E(k)$.
According to Lemma 10 of this…
Fred Akalin
- 869
2
votes
3 answers
How to do concrete calculations in algebraic geometry?
I'm working through Shafarevich's Basic Algebraic Geometry, (the first edition, with all three parts in one book), and although I've been able to deal with the conceptual problems or those proving certain varieties are (not) isomorphic/birational,…
littleman
- 454
2
votes
1 answer
Isomorphic affine classical varieties over a non-algebraically closed field. Does the isomorphism preserve polynomials?
$\def\bbA{\mathbb{A}}
\def\spec{\operatorname{Spec}}
\def\sO{\mathcal{O}}
\def\sA{\mathcal{A}}$Let $k$ be a non-algebraically closed field. Let $X=V(I)\subset\bbA^n_k$ be a classical affine variety (where $I\subset k[x_1,\dots,x_n]$ is an ideal),…
2
votes
1 answer
Fibers of Spec maps: is $\operatorname{Spec} \Bbb C[x,y]/(x^3+x^2y+xy^2)\to\operatorname{Spec} \Bbb C[x]$ finite?
Let $R = \mathbb{C}[x,y]/(x^3+x^2y+xy^2)$. Look at $R$ as an $\mathbb{C}[x]$ algebra via the trivial map (I presume $\mathbb{C}[x] \to R$ with $x\mapsto x$). Is this a finite algebra? Translate the algebra into a geometric map
$$ V(x^3+x^2y+xy^2)…
user1072285
- 681
1
vote
0 answers
Does Fulton's proof of Cayley-Bacharach Theorem (two cubics that intersect a third cubic in 8 points share the 9th) require proof by contradiction?
Prop. 3 of Sec. 5.6 of Fulton's "Algebraic Curves" is:
Let $C$ be an irreducible cubic, $C'$, $C''$ cubics. Suppose $C' ⋅ C = ∑_{i=1}^9 P_i$, where the $P_i$ are simple (not necessarily distinct) points on $C$, and suppose $C'' ⋅ C = ∑_{i=1}^8 P_i…
Fred Akalin
- 869
1
vote
1 answer
Local equation of a line near a point in a cone
I was reading Shafarevich's Basic Algebraic Geometry 1, and I was having a little trouble with the concept of a local equation. I found this link: Explicit example of local equation
but I can't quite follow the answer as I don't know what divisors…
xyzsd3324
- 43
1
vote
1 answer
Restriction of a morphism to the inverse image of a closed set
Let $X$ and $Y$ be varieties and let $\varphi:X\to Y$ be a morphism. If $Z$ is a closed subset of $Y$, is it true that $\varphi|_{\varphi^{-1}(Z)}:\varphi^{-1}(Z)\to Z$ is also a morphism? My knowledge of the subject is currently limited to…
rndnr562
- 136
- 1
- 7
1
vote
1 answer
Is a morphism from a quasi-affine variety to a quasi-projective variety given by globally defined regular maps?
$\def\bbA{\mathbb{A}}
\def\bbP{\mathbb{P}}
\def\sO_{\mathcal{O}}$The following discussion is strictly classical. Throughout this question, I will use the notions of (i) sheaf of $k$-algebras, (ii) $k$-ringed space and (iii) morphism of $k$-ringed…
0
votes
2 answers
Confusions about $\mathrm{Spec}(R)$ and algebraic groups
So I am studying about classical algebraic geometry (the one working with affine varieties instead of schemes) and I suppose I'm lost in a number of points. Note that I'm trying to study affine varieties/algebraic sets so please make sure that…
ARYAAAAAN
- 1,709
0
votes
0 answers
Defining Group Law on Reducible Cubic without Multiple Components
I am having trouble with exercise 5.35 in Algebraic Curves by Fulton, in the case where $C$ is reducible without multiple components.
For context, if $C$ is an irreducible cubic over some algebraically closed field and $C^\circ$ denotes the simple…
0
votes
0 answers
Image of a dominant map contains an open set
Consider a dominant morphism of affine irreducible varieties, the image contains a non-empty open subset of the codomain.
I know that this can be proved using Chevalley's theorem, but I was wondering if there was any "elementary" proof.
Fede1618
- 47
0
votes
1 answer
How to compute function field (for a curve family)?
This might be a stupid question... I‘m not good at Algebraic Geometry...
I'm reading a paper on Algebraic geometry code. There is a family of curve named as 'Garcia-Stichtenoth Curves'.
The function field for $X_1$ is $k(x_1)$, and for $X_2$ is…
ZWJ
- 571
0
votes
0 answers
Locus where a rational map is not defined
I want to prove that given a rational map $f:\mathbb{P}^n\dashrightarrow\mathbb{P^m}$ the locus where it is not defined $X\subset \mathbb{P}^n$ is a projective variety such that any irreducible component has dimension $\operatorname{dim}\ge n-m-1$…
Samuel Ascoli
- 78
- 9