Tagged Questions

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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14 views

Family of “something very close to be a curve” over a curve $C$

Hartshorne (IMHO restrictive) definition of a curve: Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a ...
3
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0answers
20 views

Contents of Tor modules

I'm interested in knowing a concrete description of what elements of Tor modules $\mathrm{Tor}^i_R(M,N)$ "are". As it stands I have no real intuition for, say, maps between Tor modules induced by ...
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0answers
24 views

first chern class

If $M$ is a Fano manifold, and $K_M$ is the canonical line bundle of $M$. If $L$ is an ample line bundle over $M$, and $c_1(L)=\lambda c_1(M)$, for some positive number $\lambda$. What is the relation ...
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1answer
20 views

Find the equation of the ellipse with given foci and $a$

I have to find the equation of the elipse with foci: $$(-1,-1),(1,1)$$ and $a = 3$ I could do it using the definition of elipse, which I know how to work with. But I need to do it using translation ...
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1answer
35 views

Definitions of a very ample invertible sheaf

At the moment, I'm struggling with the following definitions i) and ii). I'd like to know why they are equivalent: Let $\mathcal{L}$ be an invertible sheaf on a variety X. i) $\mathcal{L}$ is ...
3
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1answer
58 views

Question about dimension of varieties: where is this hypothesis necessary?

I found the following result on the web: Theorem. Let $f:X\rightarrow Y$ be a morphism of varieties, and assume that the dimension of all fibers $n=f^{-1}(P)$ is the same for all $P\in Y$. Then $\dim ...
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0answers
16 views

Is a form of a linear algebraic group over $k$ a linear algebraic group over $k$?

As the title says. $k$ : field with char($k$) = $0$ form: if $G$ is a l.a.g. over $k$ a form of $G$ is an algebraic group $G'$ over $k$ such that $G_{\bar{k}} \cong G'_{\bar{k}}$.
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1answer
15 views

If $\varphi\colon X\to Y$ is regular, why is $\overline{\varphi(Z(J))}=Z((\varphi^{*})^{-1}(J))$?

I'm unsure about a basic property of regular maps. Can someone please clarify? Let $\varphi\colon X\to Y$ a regular map between quasi-affine sets. If $k$ algebraically closed, and $J$ an ideal in the ...
2
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2answers
33 views

Reference request for equality of torsion of H1 and H2

I have heard that for a surface $X$ (algebraic? smooth? compact?) the torsion part of $H_1(X,\mathbb{Z})$ is the same as that of $H_2(X,\mathbb{Z})$. Please could you give me a correct statement? I ...
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0answers
19 views

two projective lines at c^k meet at a point

We have an exercise saying that 2 distinct projective lines on c^k over the field C meet at a point I tried the following let P1 and P2 be the projective lines respectively. Then P1=[kx,ky] where k ...
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41 views

Gluing sheaves together

I am doing the following exercise 2.7D from Ravi Vakil's notes. Suppose $X = \cup U_i$ is an open cover of $X$ and we have sheave $F_i$ on $U_i$ with the following isomorphisms $\phi_{ij}:U_i ...
2
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0answers
65 views

Isolating x and z in two equations.

I am working on a computer program and at some point I need to isolate an x and a z. I am basically trying to isolate x and z in these two equations: 1) $xn_{x} + yn_{y} + zn_{z} = n_{d}$ 2) ...
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0answers
24 views

Harnack's curve theorem for zero sets of real trig polynomials in two variables?

Is there a result like Harnack's curve theorem for real trig polynomials which gives bounds for the number of connected components the zero set can have in terms of the degree? More specifically, let ...
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31 views

Can you help with the below as i am stuck [on hold]

We now consider the cost of manufacturing the articles. Suppose that the relationship between price and demand is still and that the cost £C thousand of manufacturing d thousand is given by . The ...
3
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2answers
45 views

The arithmetic genus of non-reduced curves

Let $(X,h)$ be a smooth projective variety, and let $C\subset X$ be a smooth rational curve. Then $C$ has arithmetic genus $0$. (That $p_a(C)=0$ is not important, just to fix ideas). But if I am ...
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3answers
28 views

basic question regarding the definition of sheaf of rings

I was wondering and got confused about something. Say we have a sheaf of rings $F$ on a topological space $X$. Let $U$ be an open set of $X$, then by the definition $F(U)$ is a ring. I was wondering, ...
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1answer
21 views

Does the sign of $a(x-x_0)+b(y-y_0)+c(z-z_0)$ represent which side of the plane $M$ is?

For $P(x_0, y_0, z_0)$ on a plane of normal vector $N(a, b, c)$ and a random point $M(x, y, z)$ does the sign of $a(x-x_0)+b(y-y_0)+c(z-z_0)$ represent which side of the plane $M$ is on (and is null ...
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0answers
20 views

Definition of multiplicity of intersection

I'm reading this paper and I don't know this definition in page 3: What is the definition of multiplicity of the intersection of a hyperplane $H$ at a point $P$ in a curve $X$? Remark: My only ...
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0answers
32 views

Irreducible Quasi-affine Varieties and Dominant Maps

Problem Definition: A quasi-affine variety is a topological space $X$ with an algebra of functions $\mathcal{O}(X)$ such that there is a homeomorphism $\psi:X\to U$ for some open subset $U$ of ...
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0answers
61 views

Except First year Abstract Algebra and commutative Algebra, what else do i need to start read Algebraic Geometry text?

Except First year Abstract Algebra and commutative Algebra text, what else do i need to read before start read Algebraic Geometry texts? I am refer to the beginning texts: "Algebraic geometry an ...
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0answers
7 views

Parametrization of quadratic (diophantine) equations with a nth power

Is it always the case that the general solutions can be readily found if the primitive ones are known? If so, can this be applied to $ax^2+by^2=cz^n$ if the primitive solutions of $x^2+y^2=c^r$ are ...
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1answer
20 views

On Rational function fields of varieties

It is given that $Q=V(x^3-yz^2)$ is a variety in $\mathbb P^2$ (i.e. projective space of dimension two).Then what is the rational function field K(Q) of Q? 2.Consider the hypersurface ...
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1answer
37 views

Graph on complex plane $x^2+y^2=1$

Our professor told us that any affine plane curve in complex plane is unbounded so I am wondering what is the graph of $$x^2+y^2-1=0$$ Is it not a circle?
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1answer
50 views

When does Coh$(X)$ have enough locally frees?

Let $X$ be a scheme of finite type over $\mathbb C$. One might be interested in morphisms in the derived category $D(X)$ of coherent sheaves on $X$, that are morphisms $f:E^\bullet\to F^\bullet$ of ...
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1answer
23 views

Subgroups of isometries on Euclidean space

I am solving the following exercise: Let $\mathcal{T}(E) := \{ T_v \mid v \in \mathbb{R^2} \}$ be the set of all translations of E and $\mathcal{O}(2,\mathbb{R}) := \{g \in ...
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1answer
60 views

A question from Algebraic Geometry: scheme over the spectrum of a DVR

I am trying to read a paper entitled " Valuative Criteria for Families of Vector Bundles on Algebraic Varieties" by Stacy G. Langton. There, the author mentions the following- Let $(R, m ,k)$ be a ...
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1answer
35 views

Identifying the conic $x^2 -2xy +y^2 -5\sqrt{2}x+3\sqrt{2}y+10 = 0$

I have to identify which conic is: $$x^2 -2xy +y^2 -5\sqrt{2}x+3\sqrt{2}y+10 = 0$$ The method my book uses is by translation and rotation. It basically proves that if we want to translate na ...
1
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1answer
27 views

conics isomorphic to $\mathbb{P}^1$ over $\mathrm{Spec}(\mathbb{Z})$

Let $C \subset \mathbb{P}^2$ be a smooth non-sigular conic over $\mathbb{Q}$, say given by an equation $f(x, y, z)=0$, where $f$ is a homogeneous polynomial of degree two with rational coefficients. ...
3
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1answer
49 views

Morphisms of quasi-affine varieties and locally closed sub-varieties

The Problem Let $\varphi:X\to Y$ be a morphism of quasi-affine varieties. Let $Z\subset X$ be a locally closed sub-variety (that is, $Z$ is an open sub variety of a closed subvariety). Show that ...
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1answer
45 views

A few questions about Mumford's Red Book

I'm studying algebraic geometry following the Red Book, and a few questions arise in the section about Dimension, that after much try I could not understand. First question (about theorem 2 of page ...
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2answers
49 views

solve this simple equation:$ax^2+byx+c=0$

I need help solving the diophantine equation:$$ax^2+bxy+c=0$$ The quadratic formula does not seem to help much. Please help.
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0answers
22 views

how do you find the parametrization of $ax^2+by^2=z^r$?

how do you find the parametrization of $ax^2+by^2=z^r$ if a non-trivial solution $(x_0, y_0, z_0)$ is known? Any hint?
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0answers
40 views

Exceptional coherent sheaves on $\mathbb{P}^n$ are vector bundles

Let $E$ be a coherent sheaf over $\mathbb{P}^n_k$. Coherent sheaf is called exceptional if $\operatorname{Hom}(E,E) \cong k$, and $\operatorname{Ext}^{> 0}(E,E) \cong 0$. How one can show that such ...
2
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1answer
28 views

Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
2
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1answer
18 views

Transformations from n-sphere coordinates to cartesian coordinates.

I was wondering how one would proceed to convert between coordinate systems in $ \mathbb R^n $. For $ \mathbb R^2 $ the conversion is easy and just basic trigonometry. Given $(r, \theta)$ we can ...
2
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1answer
53 views

Computing these multiplicities

I'm trying to use some Algebraic Geometry techniques to check my understanding on them. I'm using the most stupid of all the examples: trying to compute the multiplicities of the intersections of the ...
0
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1answer
16 views

Converse of the implication $V(S)\subseteq V(T)\iff T\subseteq\sqrt{\langle S\rangle}$.

I'm having trouble recalling one direction of the following bi-implication. Suppose $S,T$ are subsets of the polynomial ring $k[X_1,\dots,X_n]$ over an algebraically closed field. We have ...
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0answers
50 views
+50

Rational points of $ax^2+by^2=z^r$, $r $ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
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1answer
50 views

Geometric meaning of intersection multiplicities?

I am wondering about the geometric significance of the intersection multiplicity of two curves as defined in Hartshorne 5.4 (The length of $O_p/(f,g)$ is the intersection multiplicity of $Z(f)$ and ...
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0answers
34 views

Elimination theory in Hartshorne

Does anyone know a good reference for elimination theory (Theorem 5.7A) mentioned in Hartshorne? The reference he gives is Van der Waerden modern algebra volume two, but it didn't feel locally ...
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0answers
33 views

about the spanned divisor of a complex variety

I have this definition: let $\xi \in H^1(X,O^*)$ a cocycle. We say that $\xi$ is spanned if for every point $x$ in my variety $X$ there exist a section $s \in H^0(X,O(\xi))$ such that $s(x) \neq 0$. ...
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1answer
28 views

are base change and restriction of scalars “inverses” in this case?

Let $l/k$ be a finite extension of fields. Let $G$ be an affine $k$-groups scheme. Let $G_l = G \times_k l$. Is it true that $\mathfrak{R}_{l/k}(G_l) = G$, where $\mathfrak{R}_{l/k}(-)$ is the ...
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1answer
42 views

When the fibers of a flat morphism are varieties.

Notations: As in Hatshorne's book. Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field. Are the fibers of $f$ ...
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0answers
28 views

Perimeter of an ellipse intuition help

I am aware that you can take the circumference of an ellipse using an elliptic integral and haven't looked much into it, but that seems to be a bit extreme and i was taking a personal look at conic ...
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0answers
24 views

$X\to \textrm{End}(O_{X,e}/m_{X,e}^r)$ is a morphism

Suppose $X^n$ is a complete group variety over algebaically closed field $k$, then the group law can be shown to be commutative. In proving this, one step is to show $X\to ...
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1answer
16 views

Intersection of open affines in a prevariety

What is an example of a prevariety in which the intersection of some two open affines is not an open affine? My examples of prevarieties that are not varieties does not extend beyond the affine line ...
4
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2answers
97 views

Etymology of “flabby” or “flasque” sheaf

I just started working with flasque, or flabby sheaves, that is sheaves whose restriction maps are surjective for any two open set of the space. I wonder about the etymology of the term. In French, ...
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2answers
51 views

Why aren't those Cartier Divisors equivalent?

Please refer to Gathmann's notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf at Example 9.3.6 for context. It's trying to give an example that the map between $Div(X)$ and ...
1
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0answers
47 views

Intersection of Segre variety with linear spaces

Consider the intersection of the Segre variety associated to product of $n$ copies of $\mathbb P^2$, with $k$ linearly independent hyperplanes. Is it possible to drop one of the hyperplanes and obtain ...
1
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1answer
28 views

Iitaka fibration over canonical model

I am looking for a referrence for the proof of following fact If the minimal projective manifold has positive Kodaira dimension and it is not of general type, it admits an Iitaka fibration over ...