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)

1
vote
1answer
45 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
0
votes
1answer
50 views

How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
2
votes
1answer
68 views

On the definition of a reductive group.

Wikipedia defines a reductive group $G$ as an algebraic group with trivial unipotent radical. The radical is the connected component of identity in the maximal normal solvable subgroup of $G$. The ...
0
votes
0answers
21 views

Complex conics as a Riemann surface

Consider the complex curve defined by $\{(x,y) \in \hat{\mathbb{C}}^2 | ax^2 + bxy + cy^2 + dx + ey + f = 0 \}$ for some complex numbers $a,b,c,d,e,f$ (here $\hat{\mathbb{C}}$ is the Riemann sphere). ...
2
votes
1answer
31 views

Is the product of algebraic groups the same as the fibre product?

Assuming we have two algebraic groups $G_1$ and $G_2$ over $k$. Then the direct product $G_1 \times G_2$ with the direct product group structure is an algebraic group. Is this the same as the fibre ...
3
votes
1answer
79 views

Reference request: Galois descent

What is a classic (perhaps even original) reference for Galois descent? I know that it can be seen as a special case of faithfully flat descent (for which FGA and SGA I is the usual reference) and ...
3
votes
0answers
42 views

How are varieties related polynomials?

My teacher says that varieties and ideals are related to each other while I tend to mix polynomials and varieties in my terminology. Could some explain how varieties are related to polynomials? And ...
0
votes
0answers
16 views

Duality between cut ideals and cycle ideals?

There exist a general duality between vertex-cuts and cycles and also Duality Principle on Digraphs. I am trying to find a duality prienciple expressed in terms of ideals so Does there exist a ...
1
vote
1answer
63 views

Moduli Space of Hyperelliptic Curves as Fibration?

Basically, I had a thought about a way to think of the moduli space of hyperelliptic curves. I'm sure it's wrong most likely, but I was hoping someone could maybe point out the flaw in my reasoning. ...
1
vote
0answers
24 views

Example of an monomial ideal that is weakly reverse lexicographic but not reverse lexicographic

We are looking at a paper titled "Generic Ideals and Moreno-Socias Conjecture" by Edith Aguirre, et al. In the paper they state that an ideal which is reverse lexicographic is also weakly ...
3
votes
1answer
60 views

Example $3.3.1$ in Hartshorne

Let $k$ be an algebraically closed field, and let $$X = \operatorname{Spec} k[x,y,t]/(ty-x^2)$$ $$Y = \operatorname{Spec} k[t]$$ Hartshorne comments that both schemes $X$ and $Y$ are of finite type ...
0
votes
2answers
51 views

If there is a inclusion of restrictions, then is there an inclusion of sheaves?

Let $X$ be the projective space over $\mathbb{C}$. Let $H$ be a smooth hyperplane in $X$. Let $F$ and $G$ be torsion-free sheaves on $X$ of rank 1 and 2 respectively such that we have an inclusion $F|...
5
votes
1answer
71 views

Splitting a short exact sequence of complexes of vector spaces

It's well-known that any complex of vector spaces is isomorphic to a direct sum of two types of indecomposable complexes (a one-dimensional space concentrated in one degree, or two one dimensional ...
5
votes
0answers
63 views

Definition of cotangent and conormal bundle

I have read the following definition of cotangent bundle: Let $X$ be a $n$-dimensional smooth algebraic variety. For any $p\in X$ there exist a neighbourhood $U_{p}\subseteq X$ and functions (...
0
votes
0answers
48 views

Geometric and arithmetic Frobenius

I read in Serre's "Lectures on $N_X(p)$" that when $X$ is a scheme defined over $\mathbb{F}_q$ (a finite field), the geometric Frobenius $F: X \mapsto X$ is defined by fixing every element of the ...
0
votes
0answers
47 views

Kernel of the evaluation map of a sheaf has global sections?

Suppose that $X$ is a smooth projective variety over an algebraically closed field, for example $\mathbb{C}$. Let $F$ be a coherent sheaf on $X$. Consider a subspace $V$ of $\Gamma(X,F)$, the space of ...
1
vote
1answer
102 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
1
vote
0answers
36 views

Extending a morphism to a proper scheme using the valuative crierion for properness

Here's an example I'm trying to work out. Let $f : U := \mathbb{A}^1 -\{0\} \rightarrow X$ be a morphism of schemes, where X is a proper scheme over a field $k$. I am trying to extend this morphism to ...
1
vote
1answer
45 views

Global section defines a map from structure sheaf

Let $X$ be a smooth projective scheme over an algebraically closed field. Let $F$ be a coherent torsion-free sheaf on $X$. A global section $f$ of $F$ defines a morphism $O_X\rightarrow F$ given by: ...
0
votes
0answers
50 views

Underlying topological space of $X_y = X \times_Y Spec \hspace{0.5mm} k(y)$.

Let $f: X \to Y$ be a morphism of schemes, and let $y \in Y$ be a point, $k(y)$ the residue field of $y$, $Spec \hspace{0.5mm} k(y) \to Y$ the natural morphism. Let $X_y = X \times_Y Spec \hspace{0....
2
votes
0answers
33 views

Ext group of bundles on moduli space of curves

Let $\mathcal{M}_{g}$ be the moduli space of curves of genus $g$. Let's suppose $g \geq 2$. Let $T$ be the tangent bundle of $\mathcal{M}_{g}$. Is the Ext group $\text{Ext}^1(\bigwedge^2T, T)$ trivial?...
0
votes
0answers
7 views

Planar ternary ring point operations

I have the following topic in my exam questions' list: Prove that point operations in a planar ternary ring satisfy field axioms. I know Proposition 1 from this paper but this only says something ...
1
vote
1answer
23 views

Endomorphism ring of an abelian variety and its reduction mod $\mathfrak{p}$

Let $A$ be an abelian variety defined over a number field $K$. Let $\mathfrak{p}$ be a prime of $K$ for which $A$ has good reduction and let $k=\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. Let $\...
0
votes
0answers
41 views

Pullback of push forward of a sheaf

Let $X$ be a smooth projective variety over an algebraically closed field. Let $f:Y\hookrightarrow X$ be a smooth closed subvariety of $X$. Let $A$ be torsion-free sheaf on $Y$. Then consider $f^*f_*A$...
1
vote
1answer
87 views

For each $\alpha>0$ there exists a zero dimensional ideal $I$ such that $\dim_K(R/I) - |V(I)| \geq \alpha$

Let $I \subset K[x_1,\dots,x_n]$ be a zero dimensional ideal and $$V(I) = \{ (\alpha_1,\dots,\alpha_n) \in K^n: f((\alpha_1,\dots,\alpha_n)) = 0\ \forall f\in I\}$$ (the variety). I know that $|V(...
5
votes
1answer
127 views

Noether normalization in algebraically closed field

The Noether normalization lemma states that if $k$ is a field, and $A$ a finitely generated $k$-algebra, then there exist elements $y_1,...,y_m\in A$ such that $y_1,...,y_m$ are algebraically ...
2
votes
1answer
44 views

Exactness of Hom functor for torus representations?

Given a reductive algebraic group $G$ and a maximal torus $T$. Is it true that the functors $$ Hom_T(-,\lambda) $$ are exact, where $\lambda$ denotes one of the the simple one-dimensional ...
0
votes
1answer
45 views

Definition of singular points on an algebraic curve

From what I understood, given a point $p$ on a scheme $X$ over a field $k$, we have \begin{equation} \dim \mathcal{O}_{X,p} \leq \dim_{\mathcal{O}_{X,p}/\mathfrak{m} }\mathfrak{m}/\mathfrak{m}^2 \end{...
0
votes
0answers
25 views

Poncelet's closure theorem

Need some help understanding the proof made by Kneebone and Semple in "Algebraic Projective Geometry". I loose it in the sentence about the (2,2) correspondance. As I understand it, they setup an ...
2
votes
0answers
39 views

Do finite groups act admissibly on separated schemes of finite type over k

Background: Recall from SGAI that a group $G$ acts admissibly on a scheme $X$ if the quotient $X \to X/G$ exists and is an affine morphism of schemes. This is the case if and only if every orbit of $G$...
1
vote
1answer
50 views

How to represent real algebraic numbers with period integrals

Background: A real period is defined to be the value of an integral of the form $$\int_D R(x_1,\cdots,x_n)dV$$ where $R$ is a rational function with rational ceofficients, and $D\subseteq\Bbb R^n$ ...
0
votes
0answers
62 views

Metric transformation

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
0
votes
0answers
63 views

Functor associating to graded module a quasi coherent sheaf (full?)

For a graded ring $A$ there is the functor $$ ^\tilde{}\colon grA-Mod \longrightarrow ProjA-QCoh, $$ which is not faithful. I am curious if this functor is full in general or at least if we assume ...
1
vote
2answers
54 views

Residue field of the integral closure of a local ring in its field of fractions

When considering the discrete valuation rings contained in the rational functions field $R(F)$ of an irreducible plane projective curve $F \in \mathbb{P}^2(K)$ ($K$ algebraically closed), one can find ...
0
votes
1answer
51 views

Generization gives ring map

Let $X$ be a scheme, $x \in X$, and let $\tilde x$ be a generization of $x$. In other words, $x \in \overline{\{\tilde x\}}.$ I am having trouble understanding why this makes $\mathcal{O}_{X,\tilde x}...
0
votes
1answer
49 views

Pullback of global sections?

Let $X$, $Y$ be abelian varieties and let $f:X\to Y$ be a morphism. They told me that we can define the pullback $f^*s$ of a global section $s\in\Gamma (L)$ where $L$ is an ample line bundle on $Y$, ...
1
vote
0answers
90 views

Does this proof (Lie-Kolchin) suffer from a loss of injectivity?

In the following proof (after "But there is a more elementary proof"), I was confused on something. Apparently we can assume without loss of generality that $V = V_{\chi}$. In this case, here is ...
7
votes
0answers
184 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
1
vote
1answer
68 views

Coherent sheaves with no cohomology over a hypersurface

Let $X_d \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree $d$. How one can describe all coherent sheaves on $X_d$ with no cohomology i.e. $$ H^i(X_d, F) \cong 0, $$ for all $i \in \mathbb{...
0
votes
1answer
38 views

A question on very ample line bundle on Abelian Varities

I have a problem with some consideration that Mumford does about very ample line bundles in the prove of Riemann-Roch theorem. Namely, he says that if we consider a very ample line bundle $L=O(D)$ on ...
1
vote
1answer
31 views

Degree of filtered vector bundle

Suppose I have the sheaf $\mathscr{M}$ defined by $$0\to \mathscr{M}\to \mathscr{O}_{\mathbf{P}^r}^{r+1}\to \mathscr{O}_{\mathbf{P}^r}(1)\to 0 $$ that is, $\mathscr{M}\simeq \Omega_{\mathbf{P}^r}^1(1)...
2
votes
0answers
71 views

What is the geometrical interpretation of Cartier Divisors?

Definition: Let $(s, \mathcal{L})$ be a pair where $s$ is a rational section of the line bundle $\mathcal{L}$. The Cartier divisor is defined as this pair $(s, \mathcal{L})$. My question: What is ...
1
vote
0answers
32 views

Using Division Algorithm on Polynomials in Finite Field

From Ideals, Varieties, and Algorithms - Cox, Little, O'Shea. Chapter 1, Section 4. Ideals, Exercise 13 (b). Show that every $f \in \mathbb{F}_{2}[x,y]$ can be written as $f = A(x^2-x) + B(y^2-y)...
2
votes
0answers
130 views

Does Nagata theorem hold in a field that is not algebraically closed?

Let $R$ be a finitely generated $k$ - algebra and $G$ be a reductive group acting rationally on it. Then a theorem of Nagata says that the invariant ring $R^G$ is also finitely generated. Here $k$ ...
0
votes
1answer
58 views

Categorical Quotients and Group Actions on Varities

So I am given that Let $G = Z/dZ$ where d ≥ 1. Let w be a generator for G and let G act on $A^ {n+1}$ via $w(x_{0}, . . . , x_{n})$ = $(wx_{0}, . . . , wx_{n})$. How can I Show that the ...
2
votes
1answer
66 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
2
votes
0answers
21 views

Is $\overline {G^\circ\cdot p}$ a toric variety?

Consider the algebraic torus $(\mathbb C^*)^n$. Let $G$ be a subgroup of $(\mathbb C^*)^n$ that is also a reductive group. Let $G^\circ$ be the connected component of $G$ containing the identity ...
1
vote
0answers
54 views

Self-extensions of a skyscraper sheaf: algebra structure

Let $V$ be a smooth variety over a field $k$. For a point $x \in V$ we denote the skyscraper sheaf of length 1 by $$ k(x) = \mathcal{O}_x/m_x. $$ Then by taking the Koszul resolution of $k(x)$ one ...
-1
votes
2answers
98 views

Prove that $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ is an integral domain [closed]

Let $R = K\langle x,y,z\rangle/\langle x^2 - yz\rangle$ be an analytic algebra. I am trying to prove that $R$ is an integral domain. Basically I know that if $\langle x^2 - yz\rangle$ is a prime ...
0
votes
0answers
23 views

Implicitization Problem on Graphs?

I learnt the implicitization problem for varieties in introduction course on Algebraic Geometry. I am trying to understand how to formulate a similar implicitization problem on graphs where the ...