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)

3
votes
1answer
297 views

A doubt in the proof of Prop. 1.10 of Hartshorne's Algebraic Geometry

I have a doubt in the proof of Proposition 1.10 of Hartshorne's book Algebraic Geometry, which states that if $Y$ is a quasi-affine variety, then its dimension is the dimension of its closure. In ...
1
vote
1answer
94 views

Geometric meaning of being integrally closed in some overring

The geometric counterpart of integrally closed rings (in their fraction fields) are normal varieties, as described in this MathOverflow post. Is their a similar notion in algebraic geometry for being ...
1
vote
0answers
91 views

Hironaka 1964 theorem in the context of S. Watanabe 2009 book

I am trying to read the following book of S. Watanabe: "Algebraic Geometry and Statistical Learning Theory". More particularly, I am currently interested in chapter 2 and Hironaka (1964) theorem on ...
3
votes
1answer
126 views

An example of ample sheaf with no global section

In viewing the tags about ample bundle with no global sections I found an example below: If $C$ is a curve of genus $2$, and $p,q,r$ are general points on $C$, then the bundle ...
1
vote
1answer
113 views

Pushforward of pullback of an etale sheaf

I hope this question is not too elementary, but I'm a bit lost : Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the ...
4
votes
0answers
67 views

Triple Cover of the Riemann Sphere

I have the triple branched covering $X$ of $\mathbb{P}^{1}$ defined by $y^{3}=x^{6}-1$. I want to show the following: (i) The canonical embedding $\phi: X \rightarrow \mathbb{P}^{3}$ can be given in ...
2
votes
1answer
91 views

An easy proof that $\mathrm{SL}(n,F)$ is irreducible in the Zariski topology

Let $F$ be an infinite field (that is not necessarily algebraically closed) and consider the algebraic variety $\mathrm{SL}(n,F)=\mathcal{V}(\det-1)$ of $F^{n^2}$, where $$\mathcal{V}(S)=\{\alpha\in ...
3
votes
0answers
59 views

Inequality involving multiplicities of points introduced via Quadratic Transformations of a Plane Curve

I've been learning about the resolution of singularities for plane curves, and have become stuck at exercise 7.15 of Fulton's Algebraic Curves (page 91 of the PDF). The question is: Let $F=F_1, ...
4
votes
1answer
139 views

meromorphic functions on proper varieties are rational

Suppose $X$ is a proper variety over $\mathbb{C}$, is every meromorphic function rational? In the case of projective variety, can this be derived from Chow lemma? How does the GAGA principal ...
1
vote
1answer
49 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
2
votes
1answer
140 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
6
votes
2answers
105 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
1
vote
2answers
146 views

Looking for an introductory Algebraic Geometry book

I am looking for recommendations on an AG text to work through this summer, possibly with the help of a mentor. I would want this book to have some introduction to categories, and then develop the ...
3
votes
0answers
149 views

Computing the genus in positive characteristic

That's an exercise but I'm not so sure how to approach this. Let $k$ be a field of characteristic $p$ and let $f(t)$ be a polynomial in $k[t]$ of degree $d$. Let $C$ be the curve that corresponds to ...
1
vote
1answer
54 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
3
votes
0answers
116 views

Zariski closures exercise.

Compute the Zariski clousures $\overline{S} \subset \mathbb{A}^2(\mathbb{Q})$ of the following subsets: (a) $S=\{(n^2,n^3):n \in \mathbb{N}\}\subset \mathbb{A}^2(\mathbb{Q})$; (b) $S=\{(x,y): ...
4
votes
1answer
77 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
1
vote
0answers
75 views

Criterion to decide the invertibility of polynomial maps

Consider a polynomial map $f:\mathbb{R}^{n-1}\to V\subset\mathbb{R}^n$ where $V$ is $n-1$-dimensional variety in $\mathbb{R}^n$. Are there any conditions on $f$ to determine whether it defines ...
0
votes
1answer
56 views

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular

Show that there is only one conic passing through the five points $[0:0:1], [0:1:0],[1:0:0],[1:1:1]$ and $[1:2:3]$. Show that it is nonsingular
2
votes
1answer
161 views

how to show that $V( Y-X^2 )$ is irreducible?

show that $V( Y-X^2 )$ is irreducible. $Y-X^2$ is an irreducible polynomial ($Y-X^2$ cann't be factored into more irreducible components). Can we conclude that $V(Y-X^2)$ is irreducible??
1
vote
0answers
77 views

Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)$ into irreducible components

Decompose $V(Y^4-X^2,Y^4-X^2Y^2+XY^2-X^3)\subset A^2(C)$ into irreducible components. I tried like this: $Y^4-X^2=(Y^2-X)(Y^2+X)$ and $Y^4-X^2Y^2+XY^2-X^3=(Y+X)(Y-X)(Y^2+X)$. What should I do from ...
0
votes
1answer
126 views

An affine plane curve and intersection

Suppose $C$ is an affine plane curve and $L$ is a line in $A^2(k)$, $L \not\subset C$. Suppose $C=V(F), F \in K[X,Y]$ a polynomial of degree $n$. Show that $ L \cap C$ is a finite set of no more than ...
1
vote
1answer
59 views

$f_*(O_X)=O_Y$ and connectedness of fibers

Suppose $X\to Y $ is a morphism , under what conditions we have direct image sheaf $f_*(O_X)=O_Y$? For example, suppose $\tilde{S}\to S$ is a blow up, do we have $f_*(O_{\tilde{S}})=O_S$? ...
2
votes
2answers
189 views

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big?

Can the dimension of the Zariski tangent space of a complex curve at a singular point be arbitrarily big ? Is there a formula relating the dimension of the Zariski tangent space and the order of ...
2
votes
1answer
109 views

Proving the Existence of an Automorphism on $\mathbb{P}^{1}$

I recently came across the following problem while reading: Suppose that a compact Riemann surface $X$ has genus $g>1$. Let $\phi_{i}:X \rightarrow \mathbb{P}^{1}$ for $i=1,2$ be a pair of ...
-2
votes
1answer
183 views

The form of subrings of $k[[t]]$

I saw this question in an algebraic geometry book. I tried to solve this. But I did trivial thing, so I don't write what I did here. This is just self-studying. I want to learn how to solve. Please ...
1
vote
1answer
91 views

pullback of differential form by constant morphism

Let $G/S$ be a group scheme (i'm fine if you want to assume everything is affine). Let $$g : S \to G \in G(S)$$ be an $S$ point of $G$. We can define a morphism $$\phi_g : G \to G$$ defined at the ...
4
votes
2answers
73 views

Holomorphic functions on algebraic curves

I have been asked to solve the following problem, but I really need some help... How are the holomorphic functions $f:C\to D$, where $C,D$ are nonsingular algebraic curves of genus 1? I know that I ...
4
votes
1answer
148 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ are considered the same. Is it true? Why? I'm a beginner so please answer in detail.
1
vote
0answers
39 views

Multiplicity of an affine curve at a point same as that of its projectivization

Consider the projective curve $C=V(P)$ in $\mathbb{P}^2$ where $P(x_0,x_1,x_2)$ is an homogeneous polynomial of degree $d$. At a point $[a,b,1]$, the multiplicity of $C$ is ...
1
vote
0answers
45 views

Birational Variety

Given a polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $ defined as follows: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ This map defines a Variety ($V$) of dimension $2$ in ...
4
votes
1answer
97 views

Failure of Luroth's theorem for transcendence degree 3

Can somebody give an example which shows the failure of Luroth's theorem for transcendence degree 3 over $\mathbb{C}$
1
vote
1answer
54 views

What is the nature of this surface?

What is the nature of the surface whose equation is (it depends on $m$) $$x^2+2y^2+(m+1)z^2+2xy-2yz-2x+2y-4z+m^2+4=0$$
2
votes
0answers
54 views

Bott formula for projective bundles

For a projective space one has Bott formula to compute $h^q(ℙ^n,Ω^p(k))$, where $Ω^p(k)$ is the k-twisted sheaf of sections in the p-th power of the cotangent bundle of $ℙ^n$. I am wondering if there ...
6
votes
1answer
218 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
1
vote
1answer
92 views

Invertibility of a Polynomial map.

Given following polynomial map $f:\mathbb{R}^2\to V\subset \mathbb{R}^3 $: $$ (z_1,z_2)\mapsto (2z_1-z_2, 2z_1^2-z_2^2, 2z_1^3-z_2^3) $$ Is this map a bijection? If so, how?
2
votes
0answers
110 views

Two definitions of the Weil restriction.

Let $L/K$ be a galois extension with $G:=\mathrm{Gal}(L/K)$, $X$ a $L$-scheme. We have two definitions of the Weil restriction of $X$ : 1) If the contravariant functor $\mathrm{Res}^L_K(X) : (Sch/K) ...
0
votes
0answers
43 views

Some question on curves

Let $C$ be a nonsingular curve. In the proof of Theorem(V.2.17) of hartshorne book, I see the following statment: We have only to take maps $\mathcal{O} \rightarrow \mathcal{O}(-e)$ and ...
4
votes
1answer
276 views

Tor sheaves on schemes

I was trying to understand the definition of "Tor sheaves", but since it is defined in the derived category of sheaves of $\mathcal{O}_X$-modules and since I am not acquainted with derived categories ...
2
votes
1answer
224 views

Generalised rigidity lemma

The usual version of the "rigidity lemma" in algebraic geometry says something like this: If $U, \, V, \, W$ are algebraic varieties, with $U$ proper, and $f: U \times V \rightarrow W$ is a morphism ...
2
votes
1answer
111 views

Weak Kodaira Vanishing - Hartshorne III.7.1

In the Serre Duality section of Algebraic Geometry by Robin Hartshorne, the following exercise is posed: If $X$ is an integral projective scheme over a field $k$, prove that an ample invertible sheaf ...
1
vote
1answer
71 views

Quotient of group schemes and its rational points.

At the moment I have some difficulties in understanding the quotient of group schemes and so exact sequences. I am aware that precise answers would be difficult to be given without speaking of sheaves ...
4
votes
0answers
146 views

Incidence variety fo Grassmmanians

Let $k$ be an algebraic closed field (say, $\text{char}(k)\neq 2$), $n \in \mathbb N\setminus \{0\}$ and $G(m, n) = G(m, \mathbb P^n(k))$ the variety of Grassmmanian of $m$-dimensional linear ...
2
votes
2answers
82 views

Base change by a finite extension

Let $L/K$ be a finite extension of fields Now let $T$ and $X$ be $L$-varieties (we can take $T$ affine if you want) I would like to know if it is true that $$ Hom_L(T\times_K L,X) = Hom_L(T,X)^n $$ ...
3
votes
1answer
56 views

Two continuous functions agree on an open subset of an irreducible space.

I have the varieties $X,Y,Z$ where $X$ is complete. I have the morphism of varieties $f:X\times Y\rightarrow Z$, I have a closed subset $W\subset Y$ such that $f=g\circ\textrm{pr}_Y$ on ...
1
vote
1answer
72 views

Algebraic closure of a subfield of the field of fraction of a variety

I think that there is the following claim in Basic Algebraic geometry of Shafarevich. Let $X\to Y$ be a dominant morphism of varieties over an alebraically closed field $k$ of $char=0$. Let $\varphi: ...
1
vote
1answer
381 views

d-uple embedding

When one restricts the $d$-uple embedding $\mathbb{P}^n \hookrightarrow \mathbb{P}^N$ to $\mathbb{P}^{n-1} \hookrightarrow \mathbb{P}^n$, does this yield the $d$-uple embedding $\mathbb{P}^{n-1} ...
6
votes
0answers
301 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
6
votes
0answers
181 views

Fixed Point Involutions

In recent reading on Riemann surfaces and complex manifolds (primary Miranda with a few random finds online), I encountered the notion of involutions, in particular fixed point involutions. We recall ...
1
vote
1answer
134 views

Why are there no Dual-octonions?

In the case of quaternions, we can define the traditional quaternions setting the imaginary components equal to root negative one, the hyperbolic quaternions by using root positive one, and the dual ...