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
50 views

Curve of genus $g$ with a point removed

Let $C$ be a smooth projective curve of genus $g$. If we pick two distinct points $p,q\in C$, when are $C\setminus\{p\}$ and $C\setminus\{q\}$ isomorphic or not isomorphic? When $g = 0$, they are ...
0
votes
1answer
33 views

If $L$ splits $D=I\otimes N$ does $L$ split $I$ and $N$?

Let $D$ be a central simple division algebra over the field $F$. Let $D\sim I\otimes N$ where $I$ and $N$ are division algebras in Br$(F)$ and $\sim$ is equivalence in Br$(F)$. I am interested in this ...
3
votes
0answers
20 views

$O_S$ is the integral closure of $k[T]$ in $F$ for some embedding of $k(T)$ in $F$?

Let $F$ be a function field in one variable over a field $k$. Let $S$ be a nonempty finite subset of the set of all places of $F$. Let$$O_S = \{f \in F: \text{ord}_v(f) \ge 0 \text{ for all }X ...
3
votes
0answers
28 views

Module of vector fields reflexive.

While studying tensor fields in classical differential geometry, I found it crucial (at least for some approach) to know that for a smooth $n$-manifold $M$, its $C^{\infty}(M)$-module $\Gamma(M,T_M)$ ...
3
votes
2answers
35 views

Construction of Projective Varieties Question

I am struggling in understanding Mumford's construction of Projective Varieties. In the image I uploaded here, Are we to understand each $R_n$ as $M_n/P_n$, where $M_n:=${homogeneous polynomials in ...
2
votes
0answers
16 views

$a$ and $b$ irreducible polynomials such that $\forall u \in \mathbb{Q}[t], a|u(t^n)\iff b|u(t^n)$

A little context is in order. I was trying to find counter-examples to the following statement: $$\phi : X\rightarrow Y \;\text{injective} \Rightarrow \phi\otimes K : X\otimes_k K \rightarrow ...
2
votes
0answers
30 views

Calculating the Galois group of a covering map

Suppose $C$ is an algebraic curve and $\phi:C\rightarrow \mathbb{P}^{1}$ is a covering map of the complex projective line ramified at $\{0,1,\infty\}$ only. Suppose $\phi':C'\rightarrow ...
1
vote
0answers
45 views

dimension formula for fiber product of affine varieties

Let $X \subset \mathbb{A}^n, \, Y \subset \mathbb{A}^m, \, Z \subset \mathbb{A}^{\ell}$ be irreducible affine varieties and let $f: X \rightarrow Z, \, g: Y \rightarrow Z$ be surjective morphisms. ...
-3
votes
0answers
50 views

Vanishing set of a homogeneous polynomial containing projective lines

I have a few questions about the following exercise. Let $K$ be an algebraically closed field. Let $F \in K[X_0,...,X_N]$ be a homogeneous polynomial of degree $ m > 0 $ and let $Z = V(F)$. Let ...
1
vote
1answer
57 views

Inverse limit of blow up

Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that ...
7
votes
0answers
75 views

What is the “projective limit” of a polynomial?

Bayer and Mumford, What can be computed in algebraic geometry, reads (in part): Let $S = k[x_0, \ldots, x_n]$ be the homogeneous coordinate ring of $\mathbb{P}^n$. [. . .] Choose a ...
0
votes
1answer
13 views

About the maximum number of ordinary points on algebraic surface

http://mathworld.wolfram.com/OrdinaryDoublePoint.html I'm trying to figure out the (3) statement ( $\mu(d)\leq \frac{1}{2}(d(d-1)-3) $ ) That can't be true if the table bellow it is correct (and it ...
3
votes
0answers
36 views

Cokernel of map, function field.

Let $F$ be a function field in one variable with total constant field $k$, let $X$ be the set of all places of $F$, and let $S$ be a nonempty finite subset of $X$. We are interested in the dimension ...
3
votes
1answer
40 views

intersection of maximal ideals in a polynomial ring

Given $A=K[x_1,\dots,x_n]$ a polynomial ring on a field $K$, let $p(x)\in A$ be an element, and $M_1,\dots,M_s$ some maximal ideals. Is it true that $$\cap(M_i,p) = (\cap M_i,p)?$$ I obtained that ...
2
votes
0answers
30 views

Sections of ruled surfaces

The following questions maybe elementary, but I can't find them in the literature. Assume now everything I will write is defined over some algebraically closed field. Let $S$ be a (geometrically) ...
7
votes
1answer
397 views

Integral notation for degree homomorphism on algebraic cycles

In Fulton's Intersection Theory, he develops the notation $\int_X$ for the degree homomorphism from $A^*(X)$ to $\mathbb{Z}$, and I was wondering if there was a reason for the notation. Is this in any ...
3
votes
1answer
46 views

vanishing ideal of product of two affine varieties

Let $X \subset \mathbb{A}^n$, $Y \subset \mathbb{A}^m$ be affine varieties and let us consider them embedded in disjoint subspaces of $\mathbb{A}^{n+m}$. Let $p \in k[x_1,\dots,x_n,y_1,\dots,y_m]$ be ...
0
votes
1answer
19 views

Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if and only if $\phi$ is invertible in $\mathbb{C}[V]$.

This is an exercises in Ideals, Varieties and Algorithms by Cox et al. Let $V\subset \mathbb{C}^n$ be a nonempty variety. Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if ...
3
votes
0answers
29 views

Weighted blow-ups

I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$. In found some notes online saying that this is the surface $X$ ...
1
vote
1answer
62 views

Show that any finite set S $\subset \mathbb{P}^2$ is the zero set of two homogeneous polynomials

I'm currently working through Shafarevich - Basic Algebraic Geometry and I have the problem that I can't extend the results from $\mathbb{A}^2$ to $\mathbb{P}^2$. Could anyone help me?
2
votes
1answer
40 views

Characterization of linear system without base points

My question is really simple. Where can I find characterizations of linear system without base points? I searched on Hartshorne's book without success. Thanks
1
vote
2answers
71 views

Is an irreducible regular algebraic curve, connected?

Motivated by this MO post we ask: Assume that $H(x,y)$ is a real polynomial which is irreducible as a complex polynomial. Is it true to say that $H^{-1}(0)$ ($H:\mathbb{C}^{2} \to \mathbb{C}$) is ...
1
vote
1answer
45 views

linear systems and maps

Given a regular map $\varphi:C\to \mathbb P^n,P\mapsto \mathbb (f_0(P):f_1(P):\ldots:f_n(P))$, we can associate a linear system $|\varphi|$ in the following manner: let the divisor $D=-\min div(f_i)$ ...
0
votes
1answer
70 views

Example of a curve with this property

I'm reading Fulton's book and he defines the linear series $g_n^r$: So a curve $C$ is trigonal if it has a divisor which has a linear system $g_3^1$. I'm looking for a simple example of a trigonal ...
6
votes
2answers
140 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
2
votes
1answer
43 views

Minimal free resolution of the twisted cubic

This is exercise 13.15 in Harris' book "A First Course...". Let $X$ be the twisted cubic with ideal $I(X) = (XZ-Y^2,YW-Z^2,XZ-YW).$ Let $S(X)$ denote the homogeneous coordinate ring of $X$ and ...
0
votes
0answers
36 views

For what functions is this theorem correct?

Theorem$_0$: If $g:\mathbb{C}^k \to \mathbb{C}^k$ sends $(t a_1,...,t a_k)$ to $(t^{\alpha_1}b_1,...,t^{\alpha_k}b_k)$ for all $t\in \mathbb{C}$, the preimage of any point has size $\alpha_1 \cdots ...
3
votes
1answer
71 views

Fiber dimension theorem for locally closed sets

I want to prove (or to find a reference to) the following statement: Statement: Let $Z$ be an irreducible locally closed set (Zariski topology) of $\mathbb C^n$ and $\pi$ be a projection on the ...
10
votes
0answers
113 views

A “generalized field” with $q$ elements, when $q$ is any number?

It is well-known that if a finite field has $q \in \mathbb{N}$ elements, then $q$ is prime power and $q > 1$. However, various modification of the concept of a "field" have been made in order to ...
2
votes
0answers
79 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
3
votes
1answer
124 views

Proof verification of a weak version of Bezout's Theorem

I'd like to make sure here that my reasoning seems sound. I am working from Kirwan's book on algebraic curves. I was not totally happy with her proof of this theorem, so I wanted to see if I could ...
1
vote
1answer
40 views

Cohomology of Severi-Brauer varieties

What can be said about Galois-module structure of $l$-adic cohomology of a Severi-Brauer variety over a local field? In particular, I'm interested in the proof of the proposition given at the top of ...
0
votes
0answers
29 views

Normal cone and specialization

This question is from the Kashiwara and Schapira's book: Sheaves on Manifolds Let M be a closed submanifold of X and let S be a locally closed subset of X, prove that C$_M$(S) = ...
0
votes
0answers
31 views

Projective variety defined by a non-radical ideal.

In the context of the Exercise 5.3.D in Vakil's notes, I want to show that there are examples of a reduced graded ring $A$ and a non-radical homogeneous ideal $I$ such that $\text{Proj}(A/I)$ is a ...
0
votes
0answers
26 views

A question about the pull-back and proper pushforward functors (convolution product of perverse sheaves) on $SL(2,\mathbb C)$

Let $G = SL(2,\mathbb C)$, which is an algebraic group of type $A_1$ over $\mathbb C$. Let $B$ be a Borel subgroup of $G$. Let $X =G/B$. Then $X \cong \mathbb P^1$ and it has a stratification $X = ...
3
votes
1answer
74 views

AG on non-Noetherian rings

I must apologize beforehand as this question is pretty basic, but I can't seem to find a satisfying answer in the introduction section of the book I'm currently reading (if there is a page on here, I ...
0
votes
1answer
43 views

Normal bundle of a section of a $\mathbb{P}^1$-bundle

Let $X$ be a normal projective variety over $\mathbb{C}$ and let $\mathcal{L}$ be an ample line bundle on $X$. If we define $P=\mathbb{P}_X(\mathcal{O}_X\oplus \mathcal{L})$, then the quotient ...
2
votes
1answer
86 views

What shall I write for a reason for applying graduate school for algebraic geometry?

I'm a undergraduate applying a graduate school this year and now I'm writing a letter of self-introduction. To be honest, I don't know what exactly is algebraic geometry and I think 99% of ...
2
votes
0answers
20 views

Calculating the fan of projective $n$-space

I am reading Fulton's book on Toric Geometry, one of the exercises is to calculate the fan of projective $n$-space. I have no idea how to do this, any advice would be welcome.
1
vote
1answer
30 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
3
votes
0answers
42 views

Can we view the connected component of the Picard scheme $\text{Pic}_0(X)$ as a “kernel” of the first Chern class?

So on a curve, $\text{Pic}_0(X)$ is just the Jacobian variety, and just correspond to degree $0$ divisors. One way to extend the notion of divisors corresponding to a vector bundle is taking the first ...
1
vote
1answer
35 views

Motivic measure

Somebody can give me some good references for start to read Motivic-measure, Now I`m studing the Grothendieck Ring, and is necesary undertand something of motivic theory for my case, so I need a good ...
3
votes
0answers
22 views

Terminology for the difference of real dimension and scheme-theoretic dimension

Consider the scheme $\mathrm{Spec} \left(\mathbb{R}[x_1,\cdots,x_n]/(x_1^2+\cdots+x_n^2-a)\right)$ where $a$ is a real number. Scheme-theoretically, this has dimension $n-1$. But the dimension of the ...
1
vote
2answers
33 views

Are automorphism of $\mathbb{P}^2$ 4-transitive?

Given two set of four points, both of them not colinear, is there always $g\in Aut(\mathbb{P}^2)$ such that it sends one set two the other?
3
votes
1answer
32 views

Show that the number of points of $V(I)$ is at most $m_1m_2…m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle ...
0
votes
0answers
41 views

Question about Mumford's article

I'm reading the following article by Mumford speaking about theta characteristic. Mumford's article I'm trying to understand the definition af the quadric form $q$ on page 184. Here my questions: 1) ...
0
votes
0answers
11 views

uniqueness of the paramaters of the 2 dimensional normal cone

I have proved that all 2 dimension strongly conves rational polyhedral cones has the following normal form; $\sigma= \text{cone}(e_2,de_1-ke_2)$ Now what im trying to prove is the following; let ...
3
votes
1answer
65 views

Finite flat pushforward of a constant sheaf

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also ...
0
votes
1answer
28 views

Obstruction map for Quot schemes is surjective

I am reading "Lectures on vector bundles" by Le Potier and am confused about a statement in the proof of the existence theorem on page 144, after Lemma 8.6.6. Let $X$ be a projective curve (can ...
2
votes
1answer
42 views

How to reduce cubics in the plane to a canonical form?

I watched a video from Wildberger in the Differential Geometry series ( first, or third lecture, I don't remember ) where he says the following. The general format of a cubic curve is $$a x^3 + b ...