The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.
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0answers
20 views
Real points of a complex curve
Since the "real points" of a complex curve can mean a couple of different things, bear with me while I'm annoyingly formal here.
Consider first a cubic curve $y^2 = x^3 + a x + b$. Write
$$S := \{ ...
2
votes
2answers
55 views
Scheme over S and morphisms
Quoting from Hartshorne
Let $S$ be a fixed scheme. A scheme over $S$ is a scheme $X$, together with a morphism $X \to S$. If $X$ and $Y$ are schemes over $S$, a morphism of $X$ to $Y$ as schemes ...
4
votes
1answer
53 views
Hilbert polynomial of disjoint union of lines in $\Bbb{P}^3$
Let $X$ be the disjoint union of the two lines in $\Bbb{P}^3$ given by $Z(x,y)$ and $Z(z,w)$. Letting $R = k[x,y,z,w]$, I have computed the following free resolution for the homogeneous coordinate ...
6
votes
1answer
73 views
Possible mistake in exercise in Hartshorne exercise II.2.18b
I'm trying to solve Exercise II.2.18b in Hartshorne, and I've constructed what appears to be a counterexample to its statement. Can someone tell me where I've gone wrong?
The statement is as ...
0
votes
1answer
22 views
Rank of Jacobian at a singularity
Is the following proposition true?
Proposition: Suppose $\mathbb{C}\{x_1,\ldots,x_{d_1}\}/(f_1,\ldots,f_{k_1}) \cong \mathbb{C}\{y_1,\ldots,y_{d_2}\}/(g_1,\ldots,g_{k_2})$ are isomorphic complex ...
0
votes
1answer
48 views
Prove fact about polynomial in uncountable fields
$F$-uncountable field. $I_{i}$-ideal in $F[x_{1},...,x_{n}]$
$F^{n}=\cup_{i=1}^{\infty}V(I_{i})$
$V(I_{i})\subseteq V(I_{i+1})$
Prove that $\exists k, V(I_{k})=F^{n}$
All that I've find is that ...
0
votes
1answer
21 views
Archimedean spiral: arc length of coil
Are the arc lengths of the coils - i.e. the parts 0-2pi, 2pi-4pi, etc. - in arithmetic progression?
5
votes
2answers
276 views
Quasiseparated if finitely covered by affines in appropriate way
I've been reading Vakil's notes on algebraic geometry (on my own -- this is not part of a class), and I'm stuck on one problem (number 6.1.H). It goes as follows.
Let $X$ be a scheme. Prove that ...
1
vote
1answer
36 views
When is the image of a closed point closed under a morphism between schemes?
Let $f: X \rightarrow Y$ be a morphism between schemes. When is the image of a closed point closed? In another question , some remarks were already made. For example if $X$ and $Y$ are of finite type ...
1
vote
0answers
28 views
Is the product of non-separated schemes non-separated?
My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
0
votes
1answer
53 views
Existence of projective curve such that…
Is true that for any two different integers $d,d'>1$ there exist two projective curve, of degree $d$ and $d'$ that are not isomorphic and two projective curves that are birational ?
3
votes
2answers
89 views
$\mathbb{A}^2\backslash\{(0,0)\}$ is not affine variety
In our lecture notes we have this example, with the proof why $X = \Bbb{A}^2\setminus \{(0,0)\}$ is not an affine variety:
Let $i:X\hookrightarrow \mathbb{A}^2$ be an inclusion map. We show, that any ...
3
votes
1answer
27 views
Hartshorne III.7.6b) (ii) => (i) "Duality for a projective scheme)
Let X be a closed immersion of dimension n in P = *P*$^N_k$, where k is an algebraically closed field. Let $\omega_P$ denote the canonical bundle and A the local ring $\mathcal O_{P,x}$.
Then ...
7
votes
2answers
145 views
Derived Category of Coherent Sheaves on Elliptic Curves
I know little about algebraic geometry, however while studying noncommutative geometry some results showed that a category I understand well (holomorphic vector bundles over noncommutative tori) was ...
0
votes
0answers
40 views
What is a “mere cover”?
Sorry to ask such a basic question, but I'm having a lot of trouble finding a definition of this. I saw this term in Stefan Wewers' thesis and it seemed familiar, but googling "mere cover" doesn't ...
6
votes
1answer
55 views
Quasicoherent ideal sheaves on open subschemes
Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since ...
5
votes
1answer
42 views
Why are projective spaces over a ring of different dimensions non-isomorphic?
Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
4
votes
1answer
71 views
Are these rational curves?
I have to find the singular points of the following curves and tell if they are rational. The curves are $C=Z(x^2+y^2+x^2y^2)$ and $C=Z(x^3+y^3-1)$, and the base field is the complex one.
I think I ...
1
vote
1answer
82 views
Projective Normality
What is the significance of studying projective normality of a variety ? How does it relate to non-singularity, rationality of a variety ?
1
vote
1answer
50 views
Dimension of irreducible of finite type k-scheme
I saw a claim that states for a field $k$ and an irreducible of finite type $k$ scheme $X$, $\textrm{dim}X=\textrm{dim} \mathcal{O}_{X,x}$ for any closed point $x$.
The proof starts with reducing the ...
2
votes
0answers
72 views
Gluing schemes Hartshorne example
In example 2.3.5 Hartshorne says
Let $X_1$ and $X_2$ be schemes. Let $U_1 \subseteq X_1$ and $U_2 \subseteq X_2$
be open subsets, and let $\varphi: ( U_1, \mathcal{O}_{X_1 \mid U_1}) \to ( U_2, ...
2
votes
2answers
72 views
Separated and Finite Type Scheme over an Algebraically Closed Field
Let $(X,\mathcal{O}_X)$ be a separated scheme and of finite type over an algebraically closed field $k$. The fact that $X$ is separated means that the image of $X$ under the diagonal morphism $\Delta: ...
4
votes
0answers
67 views
approximating a variety locally by a vector space
Suppose we have $m$ homogeneous equations with integer coefficients in $n$ variables and that $m >> n$. Let $x_0 \in \mathbb{C}^n$.
Question 1: is there a way to approximate the variety ...
5
votes
1answer
64 views
Is this an affine surface?
I have to prove that the surface $Z(xy^2-z^2)$ in the affine space is a rational one, where the base field is the complex one. Is it right to use the morphism $t \rightarrow (t,t,t^{3/2})$ with the ...
17
votes
2answers
219 views
Where can I find good exercises for algebraic geometry that require hard, concrete computation?
I've been studying scheme theory from Hartshorne and Qing Liu for a few months now. (For those who are not big fans of Hartshorne, I have to note that I agree with you: I use it only for exercises.) I ...
2
votes
0answers
33 views
On a question of closed property of morphism of schemes [duplicate]
Let $f: X \rightarrow Y$ be a morphism of schemes of finite type over an algebraically closed field $k$. This means $X$, $Y$ are schemes of finite type over $k$ and $f$ is a $k$ -morphism. Let $M, N$ ...
2
votes
2answers
78 views
Hartshorne proof of adjunction formula proposition II.8.20
On page 182 Hartshorne argues that $\omega_X \otimes \mathcal O_Y = \omega_Y \wedge^r (\mathcal I / \mathcal I^2)$, where Y is a nonsingular subvariety of codimension r in the nonsingular variety X ...
2
votes
0answers
36 views
Geometric meaning of the product of ideals
Consider the partial order of quasi-coherent ideals of a scheme $X$. Actually it also carries a product which is compatible with the partial order. (This makes this partial order a commutative ...
2
votes
0answers
66 views
Question on integral scheme
Let $X=\operatorname{Spec}A$ be an affine scheme. In the book of Hartshone, he claimed that $X$ is integral if and only if $A$ is an integral domain. If $X$ is integral then we can deduce easily that ...
1
vote
0answers
28 views
Theorem on the Dimension of Fibers for locally closed an open sets
I map a locally closed set $X$ on an open set $Y$ by a polynomial map $f$.
I know that for $y\in Y$, $\dim f^{-1}(y)=0$ (finite number of points).
Is it true that $\dim X = \dim Y$?
I found the ...
1
vote
1answer
24 views
Hartshorne 8.9.1 $\mathcal O_{\Delta X}$-module structure on $\mathcal I$ := the kernel of the diagonal morphism
Hartshorne asserts in 8.9.1 that $\mathcal I$, the kernel of the diagonal morphism $X \to X \times_Y X$, has a natural $\mathcal O_{\Delta X}$-module structure.
My problem is that $\mathcal I$ is an ...
1
vote
1answer
27 views
Proof convex polyhedron with line does not contain a corner if closed
The excercise I am struggling with is the following: Given a convex closed polyhedron that contains a line, the question is, whether this polyhedron can also contain a corner.
My idea was to make a ...
1
vote
1answer
50 views
Relation between spectrum of a ring and its quotient ring and localization.
Let $A$ be a commutative ring. $I$ be an ideal of $A$, $S$ be a multiplicative closed subset. We know that :
there is 1-1 correspondence between the prime ideals $\mathfrak{p}\in Spec A$ containing ...
4
votes
2answers
201 views
Relating the homogeneous coordinate ring of a projective variety with the affine coordinate ring of an affine open subset
I'm currently working on exercise 2.6 in chapter 1 of Algebraic Geometry by Hartshorne. I'm pretty confident with my answer apart from the first bit which I feel I could be "fudging". I'm looking for ...
2
votes
0answers
30 views
when is the ideal $I(\overline{V})$ of the projectivization of an affine variety generated by the homogenization of the generator of $I(V)$
Let $k$ be a field and $V \subset \mathbb{A}^n(k)$ an affine variety defined by polynomials $f_1,\cdots, f_k$. Under which conditions the ideal $I(\overline{V})$ of the projective completion ...
4
votes
2answers
73 views
Hartshorne exercise 1.6.4 : Is it true that $\mathcal{O}_{P,X} \cong \mathcal{O}_{\varphi(P),\Bbb{P}^1}$?
Let us work over a fixed algebraically closed field $k$ and consider a non-singular projective curve $X$ and $\varphi : X \to \Bbb{P}^1$ a non-constant morphism.
My question is: For $P \in ...
2
votes
1answer
72 views
Moving lemma - from Liu's book
This is from Liu's book.
Let $X$ be an irreducible quasiprojective variety over an infinite field. Let $D_1 \ldots , D_n$ where $n=\dim X$ be Cartier divisors on X. Show that there exists $D_i' \equiv ...
5
votes
2answers
93 views
Change of variables in $k$-algebras
Suppose $k$ is an algebraically closed field, and let $I$ be a proper ideal of $k[x_1, \dots, x_n]$. Does there exist an ideal $J \subseteq (x_1, \dots, x_n)$ such that $k[x_1, \dots, x_n]/I \cong ...
2
votes
0answers
40 views
How to define the natural map on the second page of a spectral sequence?
I'm learning about spectral sequences in Ravi Vakil's notes, and can't quite figure out how to define the map ($d_2$) on the bottom of page 59 (he describes it as a worthwhile exercise). It should be ...
2
votes
1answer
41 views
Formal completion as a functor
Let $X$ be a scheme with closed subscheme $Z$.
There is a natural way to think of $X$ as a functor from schemes to sets, $$X : S \mapsto X(S) = \mathrm{Mor}(S,X).$$
It seems there will be a similar ...
1
vote
0answers
28 views
Edge in a convex polytope
I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
2
votes
1answer
55 views
Confusion regarding various definitions in defining singular homology
In defining singular homology,
A singular $n$-simplex is a continuous mapping $\sigma_n$ from the
standard $n$-simplex $\Delta^n$ to a topological space $X$. Notationally,
one writes ...
-1
votes
0answers
25 views
Counting $\#\mathrm{GL}_m(\mathbb{F}_{p^n})$ [closed]
Let $\mathbb{F}_{p^n}$ be the finite field with $p^n$ elements where $p$ is a prime. Is there a formula counting the order of $\mathrm{GL}_m(\mathbb{F}_{p^n})$?
I think it amounts to count $\#D$ ...
5
votes
1answer
132 views
Ehresmann Connection of the tangential bundle & Chern classes
I must have mistunderstood something, this is giving me quite a headache. Please, do stop me once you notice an error in my thinking.
The Ehresmann Connection $v$ of some Bundle, $E\to M$, is the ...
-3
votes
0answers
34 views
Integral extension ring bodies and [closed]
Ajunda need to solve the following question:
Be a body K and D = K [X] the polynomial ring in one variable over K.
a) If f is in D but not in D then K / K [f] is the whole extent.
b) If A is a subring ...
0
votes
1answer
85 views
Algebraic geometry: an exercise
Let X be the Riemann sphere with local coordinate $z$ in one chart and $w=1/z$ in the other chart. Let $\omega$ be a meromorphic 1-form on X. Show that if $\omega=f(z)dz$ in the coordinate z, then f ...
5
votes
1answer
44 views
Fibre products of closed subschemes
Let $U = Spec(A)$ and $V = Spec(B)$ be affine schemes. Let $X$ be a separated scheme. Suppose that there exist morphisms $U \to X$ and $V \to X$. Then is the natural map $$A \otimes_{\Gamma(X, ...
3
votes
1answer
45 views
is prime spectrum $Spec(R)$ countable?
Let $R$ commutative ring with identity, given $Spec(R)=\{I|\text{$I$ prime ideal of $R$}\}$, does the set $Spec(R)$ countable? Also, if $\{\langle p^n \rangle\}$ is closed in $P_{-}Spec(R) = \{I| ...
0
votes
0answers
38 views
conditions of existence non trivial solution of two quadratic polynomials in three variable
I need to find conditions for the existence of non-trivial solutions to a multivariable polynomial system in two cases:
The first case:
$f_1$: $a_1x^2+a_2xy+a_3y^2+a_4z^2=0$
$f_2$: ...
3
votes
1answer
52 views
Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
Problem: Show every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$
Thoughts: For a (quasi-projective) variety $X,$ the function field $k(X)$ is a finitely ...
