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
1answer
52 views

Determinant of a coherent sheaf over a smooth projective variety

We know a coherent sheaf $E$ over a smooth projective variety $X$ admits a finite locally free resolution. $0\longrightarrow E_n\longrightarrow E_{n-1}\longrightarrow\cdots\longrightarrow ...
1
vote
0answers
37 views

Find the irreducible components of algebraic curve. [on hold]

Ley $Y$ be the algebraic set in $\mathbb{A}^3$ defined by the two polynomials $x^2-yz$ and $xz-x$. show that $Y$ is a union of three irreducible components. Describe them and find their prime ideals. ...
2
votes
1answer
40 views

Morphism and Composing morphisms of Varieties

Hi guys I am trying to convince myself that composition of morphisms is again a morphism. If $\phi: V \rightarrow W$ and $\psi : W \rightarrow Z$ are morphisms of varieties. Then $\psi \circ \phi : V ...
0
votes
1answer
40 views

image of Segre-Veronese as a tuple of polynomials

This question shares the same context as pullabck of rational normal curve under Segre map, but it is otherwise independent. It relates to Exercise 2.29 in Harris (AG-first course). So we begin with ...
3
votes
1answer
48 views

Maximal ideal in a polynomial ring over a field that is not algebraically closed

I want to prove that although $K$ is a field that IS NOT algebraically closed, every maximal ideal in $K[x_1, \ldots, x_n]$ can be generated by $n$ elements. To prove this, I am following the next ...
0
votes
0answers
21 views

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
3
votes
1answer
96 views

pullabck of rational normal curve under Segre map

Let $\nu:P^1 \rightarrow P^2$ be the veronese map of degree $2$, i.e. $[Y_0 : Y_1] \mapsto [Y_0^2 : Y_0 Y_1 : Y_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Consider the ...
1
vote
0answers
34 views

degrees of L-functions and dimensions of Shimura Varieties

I try to grab a (very) little understanding of what a Shimura variety is, and although I still don't understand the formal definition of that notion, a few vague ideas have come to my mind. Hence ...
1
vote
2answers
55 views

Manifold over a Finite Field

I'm trying to either associate a manifold with a finite field, or, ideally find a way of considering finite fields as manifolds, in a non-trivial manner. I hope to be able to use this to extend ...
1
vote
0answers
50 views

A good book to read with Chapter III of Neukirch's “ANT”

The book Algebraic Number Theory from Neukirch is a beautiful book in ANT, but it still have a serious lack in examples and motivation to the concepts. I've already read the first two chapters of the ...
1
vote
1answer
38 views

Vector bundles on $\mathbb{A}^1_k$ with doubled origin?

One of the most common examples of gluing affine lines is the affine line $\mathbb{A}^1_k$ with doubled origin. Out of curiousity, is there a known classfication of the vector bundles on this space?
0
votes
1answer
20 views

Polynomial approximation on affine varieties

Let $V,W \subseteq \mathbb{A}^n$ be two affine varieties over an algebraically closed field $k$ of characteristic zero and let $a,b\in k$. Q: Can we find a polynomial $f \in k[X_1,...,X_n]$ such ...
0
votes
0answers
35 views

Gieseker's theorem for surfaces of general type

I'm trying to understand what is the geometrical meaning of the Gieseker's Th. that is There exist a quasi projective moduli space for surfaces of general type with fixed invariants $K^2$ and $\chi$. ...
0
votes
1answer
32 views

Determinant of this coherent sheaf on a surface $S$

If $C$ is a curve on a surface $S$, i.e. $i:C\subset S$, and $G$ is a line bundle on $C$, then $G|_U\cong \mathcal{O}_C$ where $U$ is an open subset of $S$, that is, $G$ is trivial on the complement ...
0
votes
1answer
37 views

A torsion-free sheaf of rank 1 on a surface

Let $X$ be a surface and $E$ be coherent sheaf on $X$. Now there is always a natural map $\mu:E\longrightarrow E^{\vee\vee}$. The kernel of this map is precisely the torsion subsheaf of $E$. Now if ...
1
vote
1answer
52 views

Euler characteristic of a singular fiber

I am trying to understand Kodaira's classification of fibers. In the table at page 41 of Miranda's book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf there is given the Euler number of the ...
1
vote
1answer
32 views

How do I know an element generates a coordinate ring K[W] as a vector space over K?

I have an example which proves that a cuspidal cubic $W\subset \mathbb{A}^2$ defined by $y^2-x^3=0$ is not isomorphic to to $\mathbb{A}^1$. I'll start by defining a few things: Let $V=\mathbb{A}^1$, ...
2
votes
1answer
30 views

If $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$ are in $d$-general position, then they are in $1$-general position.

Let $\mathcal{L}_{d}^{n}$ be the $\binom{d+n}{n}-1$ dimensional projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ and $p_{1},\ldots,p_{r}\in\mathbb{P}^{n}$. We denote by ...
12
votes
1answer
314 views

Krylov-like method for solving systems of polynomials?

To iteratively solve large linear systems, many current state-of-the-art methods work by finding approximate solutions in successively larger (Krylov) subspaces. Are there similar iterative methods ...
3
votes
1answer
100 views

When a holomorphy ring is 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 ...
3
votes
0answers
74 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
0
votes
0answers
29 views

question about theorem references (who made it, year, etc.) [on hold]

The statement of the theorem that i would like to know some references is this: if we fix two numerical invariant $K^2$ and $\chi$ then there exist a quasi projective moduli space of the canonical ...
0
votes
1answer
38 views

Help to undertand the meaning of bounded family of surface

At this link article there is the Beauville's article that i'm reading for my thesis. For me it is not clear what the author means when he uses the term "bounded family" at page 124 after Theorem ...
2
votes
1answer
24 views

Inclusion of quotient sheaves restricted to open subset

When introducing sheaf cohomology following for example Chapter 8 of Kempf's book on Algebraic Varieties, we make the following standard definitions. If $\mathcal{F}$ is a sheaf of abelian groups on ...
1
vote
1answer
36 views

Computing a map in the long exact sequence (sheaf cohomology)

Let $E \subset \mathbb P^2$ be a curve cut out by a homogeneous polynomial of degree 3. This is an elliptic curve and so $H^0(E, \mathscr O) = H^1(E, \mathscr O) \cong \mathbb C$. Now I want to ...
-1
votes
0answers
75 views

Overrings of holomorphy rings

Let $F$ be a function field and $S$ be an arbitrary (and non trivial) subset of the set of places of $F$. Let $H=\bigcap_{P\in S} O_P$, where $O_P$ is the valuation ring associated to the place $P$. ...
1
vote
1answer
28 views

Decomposition of $\pi\colon E\to\mathbb{P}^1_k$ as a direct sum of tensor powers of the tautological line bundle?

Suppose you have a vector bundle $\pi\colon E\to\mathbb{P}^1_k$, where $k$ is some field. Is it always possible to decompose the vector bundle into a direct sum of tensor powers of the tautological ...
0
votes
1answer
42 views

Linear Systems And invertible Sheaf

Hello Fellow Mathematicians/Algebraic Geometer , This is question has two parts one which is more conceptual and the other more straight forward: i) Let $X$ be a non-singular protective variety ...
1
vote
1answer
26 views

Dense basic open set contained in dense open subset

For an affine variety $X$ with coordinate ring $A$ it is not hard to see that for $g\in A$ the basic open set (or distinguished open set) $$D(g):=\{ P\in X | g(P)\neq 0\}$$ is dense in $X$ if and only ...
1
vote
2answers
27 views

Approximating length of a curved line based on Begining and End points of line

I have two points, a known distance apart. At each of these points I have a sensor that gives me flow speed and direction. I originally assumed the flow path between the first point and second point ...
3
votes
2answers
300 views

Is Klein bottle an algebraic variety?

Is Klein bottle an algebraic variety? I guess no, but how to prove. How about other unorientable mainfolds? If we change to Zariski topology, which mainfold can be an algebraic variety?
2
votes
0answers
51 views

First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.

Consider a topological space $X$ and a direct limit of sheaves and morphisms $\{ \cal{F}_i, f_{ij}\}$. Define the direct limit presheaf by $U \to \varinjlim \cal{F}_i $. In general this is just a ...
1
vote
1answer
47 views

zero object in the category of group schemes

I am currently reading Ravi's lecture notes on AG, and in the introduction of group schemes(Section 6.6), he made a comment after 6.6N that the category of group schemes has a zero object. I can ...
3
votes
0answers
43 views

Axiom of glueing: direct limit of sheaves in a noetherian topological space. [duplicate]

I'm trying to prove that in a noetherian topological space the following property is satisfied: Consider a direct system of sheaves and morphisms $\{ \cal{F}_t, f_{ij} \}_t$. Consider the presheaf ...
1
vote
1answer
48 views

What is the class group of the complement of three lines in the projective plane?

I have a straightforward question : Let $ Y$ be the union of the three lines $ L_1:x=0 , L_2 :y=0$ and $L_3:z=0$ in the Projective plane $\mathbb{P }^2$. What is the Class group of the Complement ...
0
votes
0answers
29 views

Divisors on Smooth Projective Curves

Hello fellow Mathematicians/Algebraic Geometer, very straight forward questions i) Explain concretely the DVRS $R$ with $k\subset R\subset k(t)$ where $k$ is an algebraically closed field ...
1
vote
1answer
43 views

Taking module sheaf commutes with tensor product

I'm trying to prove proposition II.5.2.b in Algebraic Geometry by Hartshorne. The proposition states that for $ A $-modules $ M $ and $ N $ and $X=\text{Spec}\ A$ there is an isomorphism $ ...
0
votes
0answers
62 views

Localization of a regular local ring is regular

Quoting Hartshorne's Algebraic Geometry Definition. We say a scheme $X$ is regular in codimension one if every local ring $\mathcal{O}_x$ of $X$ of dimension one is regular. The most ...
1
vote
1answer
47 views

How to distinguish the tautological line bundle and the trivial line bundle on $P^n$?

How to distinguish the tautological line bundle and the trivial line bundle on $P^n$? How can I tell that these are not isomorphic as bundles?
0
votes
0answers
17 views

Questions about strata of a variety: the nilpotent cone of $\mathfrak{sl}_2$.

I am reading the lecture notes. In the end of page 3, let $X = \{(a, b; c, -a): a, b, c \in \mathbb{C}^3, a^2 + bc = 0\}$. It is said that there are two strata: the regular orbit $U$ and $0$. What is ...
0
votes
0answers
20 views

Trivial sections of tautological line bundle for $\mathbb{P}_F(V)$.

I have a brief question which has been bothering me. Suppose you have a finite dimensional $F$-vector space, call it $V$. Is there a nice proof of why there only exist trivial sections of the ...
1
vote
1answer
44 views

Are planes without $n$ points isomorphic as algebraic varieties for different n?

Denote $\mathbb A^d_n=\mathbb A^d \setminus \{x_1, \ldots, x_n\}$ (the algebraic variety over the field $k$). Then $\mathbb A^1_n$ are not isomorphic over $k$ for different $n$, probably because the ...
0
votes
0answers
70 views

Construction of line bundle

Let $k$ be an algebraically closed field and $C$ a smooth, projective, irreducible curve over $k$ of genus $g$. Does there exist a line bundle $\mathcal{J}$ on $C$ that has degree g and ...
0
votes
0answers
21 views

About the pluricanonical map of a surface of general type.

Reading an article by A.Beauville, i've found some facts about the pluricanonical map $\phi_{|nK_{X}|}$ of a surface $X$ of general type. For example Bombieri showed that if $n\ge5 $ then ...
8
votes
1answer
190 views

Reduction modulo p of a linear group over the rational numbers

A paper (http://arxiv.org/pdf/1407.3158v2.pdf) contains the following theorem: Suppose $\mathbb{G}$ is a connected, simply connected, semisimple algebraic group defined over $\mathbb{Q}$, and let ...
4
votes
1answer
297 views

Lines in $\mathbb{A}^3$

This seems intuitive, but I'm having trouble coming up with an exact matrix for the problem. Let $\{L_1, \ldots, L_N\}$ be a set of lines through the origin $(0,0,0)$ in the affine space ...
6
votes
1answer
157 views

Gap in Hartshorne I can't fill

Page 142, Example 6.11.4. I've been trying to go through the details of the sentence The proof of (6.10) shows that if $f \in K$ is invertible at $Z$, then the principal divisor $(f)$ on $X - Z$ ...
5
votes
0answers
54 views

Literature request: Method for constructing projective manifolds

Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by ...
2
votes
0answers
27 views

Galois action on the fibre of a morphism determined by a linear system

If $X$ is an elliptic curve, let $P,Q\in X$, then $|P+Q|$ determines a morphism $g:X\to \mathbb{P}^1$. It is easy to see $K(X)/K(\mathbb{P}^1)$ is a Galois extension of degree 2. Let $\sigma$ be the ...
0
votes
1answer
18 views

Find gradient of a equi-angular spiral (log spiral)

I encountered a problem in determining the gradient in cartesian coordinates (x,y) of a logarithmic spiral (or equi-angular spiral) profile. The log-spiral definintion is as shown below (similar to a ...