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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2
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1answer
41 views

Dimension of moduli of lines on quadric

What is the dimension of the moduli space of lines on a general quadric hypersurface in $\mathbb{P}^n$? Maybe the question is quite trivial, but different intuitive approaches (à la Italian algebraic ...
5
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0answers
45 views

How can a finite graph be viewed as a discrete analogue of a Riemann surface?

In the paper "Riemann–Roch and Abel–Jacobi theory on a finite graph" by Baker and Norine, the first line of the abstract states: "It is well known that a finite graph can be viewed, in many respects, ...
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0answers
47 views

The function field of $V=Z(y^2-x^3)$

Let $k$ be a field and let $V=Z(y^2-x^3).$ Can someone explain to me why $k(V)\cong k(s,t)$ ?? with $t=x+(y^2-x^3),s=y+(y^2-x^3)\in A(V)=k[x,y]/(y^2-x^3).$ Can we generalize it : If $V=Z(f)$ with ...
0
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1answer
22 views

maximal subtorus of a connected commutative algebraic linear group [closed]

I'm wondering the following: is the maximal subtorus of a connected commutative algebraic linear group over $k$ a) normal and closed b) defined over $k$ (for $k$ a field of characteristic zero, ...
6
votes
1answer
80 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
1
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0answers
32 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
1
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0answers
52 views

How to show $\mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme

I am reading Ravi Vakil's notes and on page 137 (June 11, 2013 ver.) he explains why $U = \mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme in the following way (Please note I am paraphrasing it ...
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0answers
21 views

Special case of Bernstein theorem

There is a Bernstein theorem which gives an estimate on the number of complex non-zero roots of system of polynomial equations. Bernstein theorem. The number of solutions in $(\mathbb{C} \setminus ...
3
votes
2answers
69 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
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0answers
47 views

What is the meaning of this notation in algebraic geometry (from /): $k\left[x_{1},\ldots,x_{r}\right]\mathbf{/\left(f_{1},\ldots,f_{r}\right)}$?

I have stumbled on something is apparently a trivial concept, but the difficulty is that I haven't seen this notation before. Here is the fragment of a text from lecture notes: Let us call $\rho$ ...
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0answers
46 views

Visualizing Line Bundles of Projective space

How should I visualize $\mathscr{O}_{{\mathbb{P}_{\mathbb{C}}}^1}(n)$?
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0answers
28 views

EGA reference for equivalent criteria for ampleness

Let $X$ be a projective scheme over a field $k$ with $\mathcal{L}$ a very ample line bundle on $X$ (very ample here means relative to the structure morphism $X \to \operatorname{Spec} k$. Where is it ...
0
votes
1answer
35 views

Quotientes of afine group schemes

Let $G=Spec(K[x_1,...,x_n])$ an afine group scheme and $H$ an subgroup scheme of $G$ then -Can i say that $H$ is a afine group scheme, if not then when can I say it? -How can I define the quotient ...
1
vote
1answer
21 views

Evaluating a function on a locally ringed space

I am just learning about locally ringed space. Let $(X, O_X)$ be a locally ringed space. It (the notes I am reading) says that: The $O_X,p$ is a local ring for each $p \in X$. Let $m_p$ be the ...
2
votes
1answer
32 views

Conic through 4 points

Let $p_1,\ p_2,\ p_3,\ p_4$ be any 4 different points on $\mathbb{CP}^1$ and $x_1,\ x_2,\ x_3,\ x_4$ are 4 different points on $\mathbb{CP}^2$. How can I show that there is unique conic $Q$ passing ...
1
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0answers
18 views

extension maps moduli space

I know that given $\phi: X \to \mathbb{P}^n$ a rational map, where $X$ is for example a projective curve, $\phi$ can always be resolved as a sequence of blowups. Now I consider the map $\phi$ that ...
1
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0answers
72 views

What are the “hidden” symmetries in Goldbach Conjecture?

What are the "hidden" symmetries in Goldbach Conjecture ? If Goldback conjecture is true, the basic instinct is that there must exist some "symmetries" which ensure (and lead) such properties. As we ...
2
votes
1answer
27 views

Problem about the kodaira's dimension

I have this exercise: Let $S$ a surface equal to the cartesian product of two projective irreducible smooth curves of genus greater or equal than 1. So $S=E \times F$. I want to describe the ...
2
votes
0answers
53 views

What are the missing gaps to prove Goldbach Conjecture?

When Andrew Wiles proved FLT, all he needed to do was to prove "semi-stable elliptic curve case" of Shimura-Taniyama conjecture. He did not need to start from scratch, he just needed to fill this ...
0
votes
0answers
21 views

what should do to start in dimension theory

I want to start reading something in dimension theory specially the defenition of a dimension of a ring and a connection between dimA and dimA[x] and depth of the prime ideal and... would you please ...
0
votes
1answer
20 views

The normalized valuation on $\mathcal{O}_{P,Y}$

Let $Y$ be a curve, and $P\in Y$ a smooth point. So $\mathcal{O}_{P,Y}$ is a regular local ring of dimension one, therefore is a discrete valuation rings. My question is what is the normalized ...
0
votes
1answer
40 views

Notation in Hartshorne book about monoidal transformations (blow-ups of surfaces along points)

I'm starting the process of learning the concept of blow-up for surfaces along a point. At page 386 of Hartshorne's book, the author defines the monoidal transformation of a surface $X$. But at the ...
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0answers
38 views

Question about Corollary I.6.6 in Hartshorne

I am having trouble understanding something in Corollary I.6.6 of Hartshorne. Let $K$ be a function field of dimension one over $k$ (by which he means a finitely generated extension of transcendence ...
3
votes
1answer
48 views

Some questions on the formation of the BSD conjecture

I'm quite curious how Birch and Swinnerton-Dyer formed their famous conjecture in the beginning of 1960s. I read some paper of Birch and Swinnerton-Dyer, as well as some paper of Tate and several ...
0
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0answers
27 views

is there an existing formula in finding the area of a rhombus wherein only the side is given?

is there an existing formula in finding the area of a rhombus wherein only the side is given? No measure of angles, no lengths of diagonals , height, etc. is given.
0
votes
1answer
45 views

Why is the multiplicative subgroup of a field an affine algebraic group?

Let $K$ be an algebraically closed field. Let $G_m$ be the multiplicative subgroup of $K$. In Lectures on Linear Algebraic Groups by Tamás Szamuely (you can easily find these notes online) it is said ...
5
votes
1answer
84 views

Bounding the cohomology of a smooth projective variety

Let $X/\mathbb C$ be a smooth projective variety. Suppose it is smoothly embedded in $\mathbf P^n$ as the zero locus of an ideal generated by homogeneous polynomials $f_1, f_2, \dots, f_r$ in $n+1$ ...
0
votes
0answers
31 views

image of the canonical morphism of a spanned divisor

Take $S$ a complex algebraic projective surface and $M\in Div(S)$ a spanned divisor of $S$. Due to the fact that $M$ is spanned we can define a morphism $\phi_{[M]} :S \rightarrow \mathbb{P}^N$ with ...
3
votes
0answers
27 views

Ample non-flex on elliptic curve

Say $E$ is a plane cubic, and $p$ is a point on $E$. Riemann-Roch tells us that $\mathcal O_E(3p)$ is very ample. If $p$ is a flex, it's easy to write down the three sections giving an embedding of ...
2
votes
1answer
55 views

When is “being a linear algebraic $k$-group” preserved?

Let $G$ be a linear algebraic group over a field $k$, with Char$(k)=0$. What "group-theoretical operations" preserve the property of "being a $k$-linear algebraic group"? For example When ...
0
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0answers
20 views

About the nef threshold of the divisor L

I have this definition: Let $S$ a complex algebraic projective surface then we can consider this set $ A=\{t\in \mathbb{R} |H+tK_S\in \overline{\mathcal{A(S)}}\}$ where $H$ is an ample divisor of ...
0
votes
2answers
37 views

Birational map and birational morphism in algebraic geometry

In algebraic geometry do the two terms "birational map" and "birational morphism" indicate the same object? By reading wikipedia the answer seems to be NO: A birational map from $X$ to $Y$ is a ...
3
votes
1answer
52 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
1
vote
2answers
67 views

How to show rational function field of an affine subvariety with dim>0 is not algebraically closed?

I do not know how to show the following statement. If $X\subset A^n$ is an irreducible subvariety, $\dim X>0$, then the rational function field of $X$, $K(X)$ is not algebraic closed. What ...
0
votes
0answers
30 views

Intersection of cylinders

Let $C \subset \mathbb{C}^3$ be a curve defined as the intersection of two algebraic cylinders: $$zP(x)+R(z)=0, \ zP(y)+R(z)=0,$$ where $P$ and $R$ are polynomials. How do I compute the genus of $C$ ...
0
votes
2answers
44 views

Multiplicity of an holomorphic map between Riemann surfaces

I need help understanding the meaning of multiplicity in a point of an holomorphic map between Riemann sufaces. So $F\colon X \to Y$ be an holomorphic, not constant map between Riemann surfaces and ...
0
votes
0answers
27 views

When is a pseudomanifold a manifold?

Under what conditions does a pseudomanifold become a manifold? I.e is there a nice conclusion we can make if our pseudomanifold has a certain homotopy type, or is possibly piecewiselinear to some ...
4
votes
0answers
55 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
0
votes
0answers
15 views

“Pseudomanifold with no singularities”

An $n$-dimensional (closed) pseudomanifold is a finite simplicial complex $X$ such that (i) every simplex is a face of an $n$-simplex (ii) every $(n-1)$-simplex is a face of exactly two ...
2
votes
1answer
57 views

Computing Images of Varieties

Somehow, this problem has been coming up a lot lately in different guises, which I'm taking as a sign that I ought to stop avoiding computational algebraic geometry. I could probably dig this up in ...
0
votes
0answers
41 views

Upper semicontinuity of fibre dimension on the target

This is Vakil 18.1.C. Suppose $\pi : X \to Y$ is a projective morphism where $Y$ is locally Noetherian (or more generally $\mathcal{O}_Y$ is coherent over itself). Show that $\{y \in Y : \dim ...
2
votes
1answer
49 views

A question regarding evaluating a function of a scheme

I am learning about schemes reading Ravi Vakil's notes. On page 136, he says "For example, consider the scheme $\mathbb{A}_k^2 = Spec \ k[x,y]$, where $k$ is a field of characteristic not $2$. Then ...
0
votes
1answer
52 views

References about algebraic geometry

My question is very simple. I'm studying a course telling about algebraic surfaces but i think that i need some knowledge about basic algebraic geometry. Do you have some suggestions?
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0answers
38 views

Comparing two definitions of determinant of coherent sheaves

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, ...
1
vote
2answers
45 views

How would I derive the equations of the family of lines on a hyperbolic paraboloid?

My textbook writes out what the equations of the two one-parameter families of lines that lie on a hyperbolic paraboloid surface are, but I am having trouble figuring out how these would have been ...
2
votes
1answer
56 views

$I(Y) = \{ p(x,y,z) \in k[x,y,z] \mid p (t,t^2,t^3) = 0, \forall t \in k \}$ is prime

I've been working on the following problem from Hartshorne: Let $Y\subseteq \mathbb{ A }^3 $ be the set $Y = \{(t,t^2 , t^3) \mid t \in k \}$. Show that $Y$ is an affine variety of dimension $1$. To ...
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votes
0answers
17 views

QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
0
votes
1answer
64 views

determining the blowup of $Y^2=X^3+X^2$ at the origin

Let $C = \{(x,y) \in \mathbb{A}^2_k : y^2 = x^3+x^2\}$ as an affine variety over some algebraically closed field $k$. From the real picture we see that this curve is self-intersecting at the origin. I ...
0
votes
1answer
28 views

what is the definition of Section

Giving a surjective morphism $\phi : S \rightarrow C$ from e complex algebraic projective surface to a projective curve i've found the term section of the morphism $\phi$ but i don't have some ...
1
vote
1answer
28 views

What about the reducible fibers of a surjective morphism?

If i have a surjective morphism $\phi :S \rightarrow C$, where $S$ is a smooth complex algebraic projective surface and $C$ a smooth projective curve, what can i say about the number of the fibers of ...