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. ...

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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 ...
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1answer
84 views
+50

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 ...
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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 ...
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1answer
25 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 $V=\mathbb{A}^1$, ...
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1answer
18 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 ...
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20 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 ...
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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
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1answer
93 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
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46 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 ...
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26 views

References to understand $K3$ surface as a double cover of $\mathbb{P}^2$ ramified along a sextic

My goal is to understand that $2:1$ cover of $\mathbb{P}^2(\mathbb{C})$ ramified along a sextic is a $K3$ surface. My main problem is in understanding the theory of ramified covering of ...
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27 views

question about theorem references (who made it, year, etc.)

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 ...
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1answer
27 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
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1answer
23 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 ...
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1answer
25 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 ...
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65 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 $P$. ...
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1answer
23 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 ...
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1answer
39 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 ...
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1answer
23 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 ...
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2answers
19 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 ...
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18 views

Find an algebraic curve passing through given points with given slopes

Let $P_1 = [X_1,Y_1,Z_1]$ and $P_2 = [X_2,Y_2,Z_2]$ be two different points in $\mathbb CP^2$ with homogeneous coordinates $[X,Y,Z]$. For simplicity, suppose that $X_1 \neq 0$, $X_2 \neq 0$ and put $u ...
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2answers
296 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?
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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 ...
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1answer
41 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 ...
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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 ...
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1answer
41 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 ...
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25 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 ...
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1answer
39 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 $ ...
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51 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 ...
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1answer
45 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?
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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 ...
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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 ...
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1answer
42 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 ...
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35 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
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46 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 ...
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17 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 ...
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1answer
151 views
+100

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 ...
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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
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1answer
151 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 $X - Z$ ...
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37 views
+50

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 ...
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25 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 ...
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1answer
17 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 ...
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20 views

Dévissage for complex manifolds

In algebraic geometry one has the following result: Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} ...
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45 views

Partial derivatives with respect to algebraically independent polynomials

Suppose that $\{f_1, \ldots, f_n\}, \{g_1, \ldots, g_n\}$ and $\{h_1, \ldots, h_n\}$ are algebraically independent polynomials that generates the same algebra of $\mathbb{R}[x_1, \ldots, x_n]$. Then I ...
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27 views

Commutative diagram of surfaces of general type

Suppose that $X$ is a complex projective surface of general type. Let $\phi_{|K_X|}$ the canonical map of $X$ and assume that the image of $X$ via the canonical map $\Sigma=\phi_{|K_X|}(X)$ is a ...
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1answer
30 views

Working with an Affine Variety and Maps

We have a $\phi :\mathbb C^4 \rightarrow\mathbb C^4$ $$\phi (a_1,a_2:b_1,b_2) = \left(\begin{array}{cc} a_1b_1 & a_1b_2 \\ a_2b_1 & a_2b_2 \end{array}\right) $$ We want to argue there exists ...
2
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1answer
71 views

Shrinking wedge of circles

I'm spending too much time thinking about this problem : I need to show that the shrinking wedge of circles which is path connected, locally path connected ,doesn't have a simply connected covering ...
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Talking about varieties

hi I was recently reading ideals varieties and algorithms. I ah having problems showing things are not affine varieties. Previously with problems like. $V= \{ (a,a) | a \in R^* \}$ it was much easier ...
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24 views

I'm having troubles to find this parametrization.

I'm reading the Reid's Undergraduate Algebraic Geometry book of algebraic geometry for undergraduates and I have two questions about a proof of an example on the page 19: Red question: Reid said ...
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1answer
111 views
+100

Help me understand Gröbner basis result please

I'm practicing a bit with Gröbner bases but I'm not understanding the following result I obtain from Mathematica: ...
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41 views

Product of schemes and ideal sheaves

Let $X \subset \mathbb{P}^n$ and $Y \subset \mathbb{P}^m$ be projective schemes over $\mathbb{C}$. Then, 1) Is the structure sheaf of $X \times_{\mathbb{C}} Y$ isomorphic to $\mathcal{O}_X ...