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. ...

learn more… | top users | synonyms (1)

0
votes
0answers
14 views

Is this equality of varieties valid?

In my book, Iam reading about the Zariski topology on varieties, however I read that the union of the varieties is again a variety and $$V(I) \cup V(J) = V(I \cap J)=V(IJ)$$ this is from Miles Reid, ...
0
votes
0answers
14 views

Questions about the Moduli Space of Vector Bundles of rank 2 with trivial determinant over a curve of genus $g$

Let $M_0$ be the moduli space of rank 2 semi-stable vector bundles over X with trivial determinant which is a singular projective variety of dimension $3g-3$. $M_0$ is constructed as the $GIT$ ...
1
vote
0answers
20 views

Transcendental solution to system of equations

Suppose $(A)$ $$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and there are functions ...
1
vote
0answers
18 views

A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
1
vote
1answer
17 views

A sufficient condition for irreducibility of a $G$-variety

Let $G$ be an algebraic group over a field $k$ and let $V$ be a variety on which $G$ acts. Suppose $U\subset V$ is a closed, irreducible, $G$-stable subset which intersects every $G$-orbit ...
1
vote
1answer
22 views

Linearly equivalent divisors and linear transformations

Let $C$ be a projective nonsingular irreducible curve. Let $D$ be a divisor on $C$. Suppose $l(D) = n > 0$ and $L(D) = \langle f_1, \ldots, f_n \rangle$. Consider the map $\varphi_D : C \to ...
1
vote
1answer
43 views

Cl(Spec A) = 0 implies A is a UFD

This is bothering me and I'd appreciate any clarification. If I know that $ \text{Cl(Spec $A$)}= 0$, why doesn't that imply that $A$ is a UFD? Why do I need that $A$ is integrally closed? This comes ...
0
votes
0answers
19 views

Showing that the image of a morphism is algebraic curve

I'm stuck at the first part. I think the image of this morphism should be described by three polynomial equations. I think I found one, which is xw-yz=0 where (x, y, z, w) is the coordinate of K^4. ...
2
votes
0answers
30 views

Computing the dualizing sheaf of an exceptional divisor

Let $D_1=Z(f_1)$ and $D_2=Z(f_2)$ be two plane cubics meeting in nine points $p_1,\dots,p_9\in\mathbb P^2$. The $f_i$'s induce a rational map $$(f_1,f_2):\mathbb P^2\dashrightarrow\mathbb P^1$$ ...
1
vote
1answer
20 views

$J \subset I(V(J))$ where $J$ is an ideal.

The textbook says it's by definition, but as I see it the inclusion should be reversed should it not? I mean $I(V(J))= \{ f\in k[x_1,\dots, x_n]: f(a_1,\dots ,a_n)=0 \text{ for an arbitrary element } ...
0
votes
0answers
31 views

Alternative proof of Noether Normalization Lemma

On Mumford's Red Book (end of section 7 of chapter 2, pg 126, 127), there is an alternative proof of Noether's Normalization Lemma that goes like this: For an affine variety $X$ over an algebraically ...
0
votes
0answers
18 views

What does it mean to restrict a function germ to a set germ?

Two sets $S$ and $T$ define the same germ at a point $\xi$ in a topological space $M$ if there is a neighbourhood $U$ of $\xi$ such that $S \cap U = T \cap U$. Two functions $f,g : M \rightarrow \Bbb ...
2
votes
1answer
24 views

Identifying lines in $\mathbb P^2$

Let $L$ and $M$ be two lines in $\mathbb P^2$. Does there exist a map $f : \mathbb P^2 \to X$ that "identifies" $L$ and $M$, in the sense that $f\vert \mathbb P^2 \setminus (L \cup M)$ is an ...
3
votes
1answer
67 views

Euler characteristic, genus and cohomology: a deep connection?

For a smooth projective curve $V$ over the complex numbers, the algebraic genus, defined as the dimension of the linear system $L(\omega)$, where $\omega$ is the canonical divisor, coincides with the ...
0
votes
0answers
21 views

Tensor product of local Artinian rings

Consider a complete Noetherian local ring $R$ and two local Artinian $R$-Algebras $A$ and $B$. I'm trying to prove that the spectrum $\text{Spec}(A\otimes_{R}B)$ is connected or, equivalently, that ...
3
votes
2answers
230 views

Why can we use flabby sheaves to define cohomology?

In my algebraic geometry class, we defined sheaf cohomology using flabby sheaves, and the functor on the category of sheaves on a space $X$: $$ D: \mathcal F \mapsto D\mathcal F $$ where $$ ...
2
votes
0answers
32 views

27 lines on Fermat cubic

Fermat cubic is $S=\{(x:y:z:w)|x^3+y^3+z^3+w^3 =0\} \in \mathbb{P}^3$. It is obvious that 27 lines on Fermat cubic are represented by $(x,ax,z,bz)$ for cube root $a,b$ of $-1$ and their conjugates. ...
2
votes
1answer
38 views

Very ample divisors and the Riemann-Roch theorem

What is the easiest way to prove that a divisor $D$ is very ample if and only if $l(D - P - Q) = l(D) - 2$ for all points $P, Q \in C$. It seems like it might be a consequence of the Riemann-Roch ...
1
vote
1answer
29 views

An algebraic curve $C$ as a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$

Could someone explain what it means for an algebraic curve $C$ to be a cover of $\mathbb{P}^1$ over $\bar{\mathbb{Q}}$, ramified over $n$ points?
0
votes
1answer
25 views

singular point of a complete intersection surface

Let $S:= H_1\bigcap H_2\bigcap \cdots \bigcap H_N \subset\mathbb{P} _{\mathbb{C}}^{N+2}$ be a complete intersection surface, where each $H_i$ is a hypersurface defined by a homogeneous equation $f_i$. ...
3
votes
1answer
33 views

Subjects or recent progress in Tropical geometry or similar suitable for undergraduate investigation

I'm worried this might not be fitting for this forum, but it's basically a literature and reference request. I'm looking to do a project in algebra where we are supposed to research some topic ...
1
vote
0answers
21 views

Invariants of $O(2) \times O(2)$ under simultaneous conjugation

Let $G= \textrm{O}(2)$ be the group of orthogonal $2 \times 2$ matrices over $\mathbb{C}$. $G$ acts on $G \times G$ by conjugation: $g \cdot (a,b) :=(g a g^{T}, g b g^T)$. This induces an action on ...
3
votes
0answers
30 views

non-abelian Galois cohomology

Let $1 \to A \to B \to C \to 1$ be a short exact sequence of (not necessarily abelian) $G$-modules. Passing to non-abelian cohomology, we have the exact sequence of pointed sets $$ 1 \to A^G \to B^G ...
1
vote
1answer
63 views

About Plücker embedding

I'm doing a work about Plücker embedding and I need some help about a few topics. I'm going to list them: $1-$ I know that Plücker embedding is well-defined and is injective. However, Plücker ...
2
votes
1answer
48 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
2
votes
1answer
46 views

Proof of Nike's trick: Two affine open subsets contain a simultaneously distinguished open subset

I'm trying to work through this proof of Nike's tick. Statement of the lemma: Let $ U_{i} = Spec\ A_{i} $ for $ i\in\{1,2\} $ be two open affine subschemes of a scheme $ X $. For $ x\in U_{1}\cap ...
0
votes
0answers
42 views

Is the image of a morphism between affine schemes always constructible?

Is there example for $f\colon A\to B$ being ring map, but the image $f^*\colon \operatorname{Spec}(B)\to \operatorname{Spec}(A)$ not constructible? (i.e., written as a finite union of locally closed ...
2
votes
1answer
48 views

$I$-smoothness in Algebraic Geometry

I was reading in Chapter 10 of Matsumura's book about $I$-smoothness. In the book, the autor defines this concept by the following universal property: Let $A$ be a ring, $B$ an $A$-algebra and $I ...
0
votes
2answers
17 views

Polynomial Ring of Linear Algebraic Group

During lectures, we defined the Linear Algebraic group as the algebraic set $ GL(V):=k^{n^2}-V(Det) $ Where $V(Det)$ are the matrices with $0$ determinant. Then we proceed by identifying the ...
0
votes
0answers
42 views

Geometric Meaning of Luna's Slice Theorem

I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful? This is Luna's Slice theorem from a ...
1
vote
1answer
33 views

Global sections of the projective space

Let $k$ be an algebraically closed field, and let $\mathbb{P}^n_k=\operatorname{Proj}(k[x_0,x_1,\dots,x_n])$, with structure sheaf $\mathcal{O}$. I would like to know how to prove that ...
1
vote
1answer
53 views

Is (0,0) of $V(x-y^2)$ a smooth point?

I'm pretty sure it is a smooth point since given $f(x,y)=x-y^2$ the gradient $df=(1,-2y)$ is always non-singular. I'm asking because page 22 of Principle of Algebraic Geometry says: ...
1
vote
0answers
45 views

Some questions about Qing Liu's proof of Valuative Criterion of Properness

I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its ...
1
vote
0answers
28 views

is configuration space an H-space?

Let $X$ be a manifold. Let $F(X,n)$ be the configuration space of order $n$. Let $B(X,n)=F(X,n)/\Sigma_n$ be the unordered configuration space of order $n$. Is $B(X,n)$ an H-space? Under what ...
1
vote
1answer
53 views

Chapter dependency tree for Hartshorne's Algebraic Geometry

I'm self-studying Hartshorne's Algebraic Geometry and I need some guidance. I've studied chapter I (varieties) and sections 1, 2 and 3 of chapter II (schemes). Do I need to study all sections in ...
0
votes
0answers
23 views

Are $k[Z_1\sqcup Z_2]\cong k[Z_1]\times k[Z_2]$ as coordinate rings?

Suppose $X$ is an affine variety over algebraically closed $k$. If you can decompose $X$ as a disjoint union $X=Z_1\sqcup Z_2$ for each $Z_i$ closed, is there some relationship between the coordinate ...
1
vote
0answers
39 views

About images of (prime) ideals under injective endomorphisms

Let $f : R \to R$ be an injective unitary endomorphism of a commutative ring with 1. Let $I$ be an ideal of $R$. I have several related questions concerning the image of $I$ under $f$: 1) Under which ...
1
vote
1answer
32 views

closed and open subscheme of affine scheme

Let $X=Spec(A)$ be a noetherian affine scheme. Let $I_1, \ldots, I_n$ be ideals of $A$ such that $I_i + I_j = 1$ for all $i \neq j$. Define $X_i = Spec(A/I_i)$ so that X is the disjoint union of the ...
0
votes
1answer
30 views

A question about varieties (proposition from Miles Reid undergraduate comm algebra)

In Miles Reid, undergraduate commutative algebra, I read the following: "Suppose that $k$ is an algebraically closed field and that $A=k[x_1,...,x_n]$ is a finitely generated $k$-algebra of form ...
2
votes
1answer
34 views

A natural map of higher direct images.

Let $f\colon X \rightarrow Y$ be a proper morphism of schemes, $Y$ noetherian (I don't know which of those assumptions is actually needed for the claim). According to my lecture notes, for ...
0
votes
0answers
28 views

Closed subscheme vs open subscheme in ring of dual numbers

If we take the ring of dual numbers $R=k[x]/(x^2)$ for algebraically closed field $k$, we note that by the ideal correspondence theorem, the only prime ideal in $R$ is $(x)$. Thus the scheme $spec ...
0
votes
0answers
35 views

History of a result from Bézout

BÉZOUT'S THEOREM: Let $F$ and $G$ be projective plane curves of degree $m$ and $n$ respectively. Assume $F$ and $G$ have no common component. Then $\displaystyle\sum_{P}I(P,F\cap G)=mn$ $I(P,F\cap ...
2
votes
1answer
34 views

Dimension of irreducible variety

Why is the dimension of intersection, $V\cap H$, of $m$-dimensional irreducible variety $V$ and a hyperplane given by $\dim(V\cap H)$ of dimension $m-1$?
1
vote
1answer
31 views

Question about composition of morphisms of schemes (Mumford's)

On section 7 of Chapter 2 of Mumford's Red Book, there is the following statement: Suppose $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$ are morphisms of schemes, such that $g$ is of finite ...
2
votes
2answers
56 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
1
vote
0answers
50 views

Is the Zariski Topology

if $ K $ is an algebraically closed field, asks: Is there a point $ "w" $ of $ K ^ n $, is closed in the Zariski toplogy?
1
vote
1answer
45 views

Is a torsor over a variety a variety?

Let $X$ be an algebraic variety over some field $k$ of characteristic 0. Let $g : Y \to X$ be a $X$-torsor under some linear algebraic $k$-group $G$. Is $Y$ also an algebraic variety over $k$?
0
votes
0answers
28 views

Galois Group acting on disjoint union of Schemes

Given a finite Galois fiedl extension $L/K$ with Galoisgroup $G$ i.e. the fixpoints of $L$ under G are $K$ ($L^G=K$). Let $G$ act on $L^n$ semilinearly (i.e. $g(ax)=g(a)g(x)$ where $a\in L$ and $x\in ...
1
vote
0answers
45 views

When is a holomorphy ring a PID?

I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. Let $F$ be a function field over a finite field $\mathbb F_q$, $S$ a non empty set of places (possibly ...
0
votes
1answer
44 views

Is quotient under $S_4$ action on “cube” representation a flat morphism?

Consider a three-dimensional irreducible representation $V$ of $S_4$, corresponding to symmetries of cube. Let $p$ be canonical projection $p: V \rightarrow V/S_4$. My question: is $p$ flat? I want ...