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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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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15 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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1answer
34 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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13 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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17 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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14 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
31 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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16 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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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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19 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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0answers
49 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
21 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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25 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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1answer
40 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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38 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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16 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
37 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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24 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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0answers
42 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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0answers
19 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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26 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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24 views

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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48 views

Intersection of valuation 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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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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44 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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0answers
36 views

Proj of some ring.

Let $R= \mathbb C[x_1,x_2,x_3,x_4,x_5,y_1,y_2,y_3,y_4,y_5]$ be the polynomial ring and let $S$ be the subalgebra generated by $x_1x_2x_3x_4x_5, x_1x_2x_3x_4y_5, \cdots ,y_1y_2y_3y_4y_5$ (the ...
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1answer
69 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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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 ...
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22 views

Finding whole number coordinates on continuous curves [on hold]

Imagine two surfaces in 3D space defined by known equations intersect and form a line in 3D. How could you find out if that curve formed by the intersection goes through any points where all 3 ...
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40 views

Motivation for Grassmannian variety

I need some information about the Grassmanian variety for my final project in algebraic geometry course that I am taking. My questions are: Why do we define the Grassmannian variety? Do we use ...
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41 views

Questions about the function fields of complex algebraic surfaces

Let $X$, $Y$ be complex algebraic surfaces(Of course, they are smooth). Suppose that $X$ is normal. Let $K(X)$ and $K(Y)$ be the function fields of $X$ and $Y$, respectively. And we have a ...
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1answer
44 views

Show that a variety is irreducible

How do I show that the variety $V = \{(x,y)\in k^2 \mid x-y=0\}$ is irreducible for an algebraically closed field $k$? One approach, I think, is to view $f(x) = x-y$ as an element in $R[x]$, where ...
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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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1answer
149 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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0answers
44 views

Does there have to be a point on elliptic curve over $\mathbb{C}(t)$

Let $E$ be an elliptic curve over $\mathbb{C} (t)$ (rational functions). I require $E$ to be defined by the following equation. $$ y^2 = x^3 + A x + B$$ Where $A, B \in \mathbb{C} (t)$. Question: ...
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60 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. $[X_0 : X_1] \mapsto [X_0^2 : X_0 X_1 : X_1^2]$ and let $\sigma: P^1 \times P^2 \rightarrow P^5$ be the Segre map. Then the image ...
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50 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
50 views

Proof of $\mathcal{O}_{\mathbb{P}^1 \times \mathbb{P}^1}(a,b)$ is ample $\iff$ $a,b >0$.

I would like some help understanding the proof in $(\impliedby)$ direction. Hartshorne on page 156, Example 7.6.2 says: If $\mathcal{L}$ is an invertible sheaf on $\mathbb{P}^1 \times \mathbb{P}^1$ ...
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2answers
37 views

Is the Godeaux surface irrational?

Studying examples of surfaces of general type, I've found the Godeaux surface. Here is a link for the definition of Godeaux's surface How can I see directly that this surface is not ruled?
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18 views

Number of fibres under finite non-flat morphism

Let $f:X \to Y$ be a finite, surjective morphism of projective, irreducible varieties over $\mathbb{C}$. We know that there is an open dense subset $U \subseteq Y$ such that every $y \in U$ has ...
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35 views
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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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2answers
23 views

Nullspace of a linear mapping from higher to lower dimension

It is a well-known fact that a linear mapping $A: \mathbb{R}^n \to \mathbb{R}^m$ , where $m < n$, has a non-trivial null space. This follows from the "rank-nullity" theorem of linear algebra. This ...
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1answer
30 views

Surface constructed using curves

Suppose that $E$ and $F$ are two complex compact Riemann surfaces with genus greater or equal than $2$. Set $$S=E \times F$$ the surface composed by the cartesian product of thees curves. What can i ...
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1answer
64 views

How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
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29 views

Subvarieties and finding ideals

Hi guys I am stuck working on this problem. I have a surface $W= V(xz-y^2)$ and we are trying to find an ideal $J \in K[W]$ so that the $V_w(J)=V(y-x^2,z-x^3)$ I showed that the second thing which is ...
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2answers
64 views

A curve has infinitely many points

Let $f\in k[x,y]$, where $k$ is an algebraically closed field. I would like to prove the curve $f(x,y)=0$ has infinitely many points. What I know is $k$ is infinite, but I don't know how to use this ...
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0answers
44 views

Is the union of two projective curves in the projective plane a projective curve?

As the title suggests, is the union of two projective curves in the projective plane a projective curve? Any help would be appreciated, thanks.
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28 views

What is the best Shafarevich's algebraic geometry book for beginners?

What is the difference between these books from Shafarevich: Basic Algebraic Geometry and Basic Algebraic Geometry 1: Varieties in Projective Space? I suppose the second one is better for beginner ...
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1answer
29 views

Is there a field k, not necessarily algebraically closed, for which variety V(y) in k^2 is reducible?

I am looking for a field $k$ such that the variety V in $k^2$ given by $V =\{(x,y)\in k^2|y=0\}$ is reducible. Thanks
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1answer
29 views

Primary Decomposition Theorem applied to projections in $\mathbb{R}^2$

We have recently learned the Primary Decomposition Theorem in my Algebra course. I came up with what I think is an instance of this theorem but I haven't convinced myself. Supposed we have the map ...