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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Question about Mumford's article

I'm reading the following article by Mumford speaking about theta characteristic. Mumford's article I'm trying to understand the definition af the quadric form $q$ on page 184. Here my questions: 1) ...
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31 views

How to compute the normal form of this geometric object?

Given this quadric: $x_1^2+5x_2^2+9x_3^2+4x_1x_2+2x_1x_3+10x_2x_3-2x_3=2$ Maple screenshots: How to put it into the normal form $\Large\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}-\frac{x_3^2}{c^2}=1$ ...
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75 views

Basis of (first de Rham) cohomology: $y^n=f(x)$

Let $K$ be a field, $f(x) \in K[x]$ be a monic polynomial with distinct roots, $\deg(f)=d$. Let $R=K[x,y]/(y^n-f(x))$ and $C=Spec(R)$. $\:\:\;\:\:\:\:\quad$ ($n>2$ integer) What is the basis ...
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20 views

Calculating the fan of projective $n$-space

I am reading Fulton's book on Toric Geometry, one of the exercises is to calculate the fan of projective $n$-space. I have no idea how to do this, any advice would be welcome.
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42 views

Can we view the connected component of the Picard scheme $\text{Pic}_0(X)$ as a “kernel” of the first Chern class?

So on a curve, $\text{Pic}_0(X)$ is just the Jacobian variety, and just correspond to degree $0$ divisors. One way to extend the notion of divisors corresponding to a vector bundle is taking the first ...
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33 views

Are automorphism of $\mathbb{P}^2$ 4-transitive?

Given two set of four points, both of them not colinear, is there always $g\in Aut(\mathbb{P}^2)$ such that it sends one set two the other?
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553 views

Class group of $k[x,y,z,w]/(xy-zw)$

I had a homework problem (II.6.5 in Hartshorne) to compute the (Weil divisor) class group of $X=\operatorname{Spec} k[x,y,z,w]/(xy-zw)$. I have accomplished this; however, I used some results I don't ...
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1answer
37 views

Motivic measure

Somebody can give me some good references for start to read Motivic-measure, Now I`m studing the Grothendieck Ring, and is necesary undertand something of motivic theory for my case, so I need a good ...
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22 views

Terminology for the difference of real dimension and scheme-theoretic dimension

Consider the scheme $\mathrm{Spec} \left(\mathbb{R}[x_1,\cdots,x_n]/(x_1^2+\cdots+x_n^2-a)\right)$ where $a$ is a real number. Scheme-theoretically, this has dimension $n-1$. But the dimension of the ...
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459 views

Sheaves created by global sections and their cohomology

I've got a question concerning sheaves created by global sections. Serre's vanishing theorem says: Let $X$ be a projective scheme over a noetherian ring $A$ with a very ample invertible sheaf ...
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78 views

Computing ${\mathcal Ext}^i_{\mathcal{O}_X}(\mathcal{O}_D, \mathcal{O}_X)$, where $D\subset X$ is a divisor

Let $X$ be a smooth scheme and $D\subset X$ be a divisor on $X$. I want to compute the sheaves ${\mathcal Ext}^i_{\mathcal{O}_X}(\mathcal{O}_D, \mathcal{O}_X)$. Actually it is quite easy. We have the ...
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1answer
66 views

Finite flat pushforward of a constant sheaf

Let $A$ be an abelian group and consider the associated constant sheaf $A$ on a (smooth projective) variety $Y$ (over a field). Let $f: Y \to X$ be a surjective finite flat morphism. Is $f_*A$ also ...
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11 views

uniqueness of the paramaters of the 2 dimensional normal cone

I have proved that all 2 dimension strongly conves rational polyhedral cones has the following normal form; $\sigma= \text{cone}(e_2,de_1-ke_2)$ Now what im trying to prove is the following; let ...
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1answer
28 views

Obstruction map for Quot schemes is surjective

I am reading "Lectures on vector bundles" by Le Potier and am confused about a statement in the proof of the existence theorem on page 144, after Lemma 8.6.6. Let $X$ be a projective curve (can ...
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24 views

Irreducible components of scheme over the 2-adic integers

Let $X=\mathrm{Spec}\,\mathbb{Z}_2[x]/\langle x^2-1\rangle$, where $\mathbb{Z}_2$ are the $2$-adic integers. What are (the coordinate rings of) the irreducible components of $X$? Here is what I've ...
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105 views

What is the importance of modules in algebraic geometry?

I have been trying to teach myself the basics of algebraic geometry. I understand the basic premise, how we define geometry spaces (algebraic sets and schemes) in terms of commutative rings. And I ...
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1answer
41 views

Number of Solutions to Polynomials in Finite Fields

Let $\mathbb{F}$ be a finite field and $f_i\in\mathbb{F}[x_1,x_2,\ldots,x_n]$ be polynomials of degree $d_i$, where $1\leq i\leq r$, such that $f_i(0,\ldots,0) = 0$ for all $i$. Show that if ...
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1answer
39 views

Calculate the angle between tangent lines on two points of a circle given a radius and a distance between them.

I want to create a formula that will calculate the angle change between two points on a circle, given the distance along the circumference of the circle between the two points, and the radius of the ...
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2answers
436 views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
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1answer
33 views

Proof that Distinguished Open Set is an Affine Variety

I'm trying to understand the proof in Mumford's Red Book that a distinguished open set of an affine variety is itself an affine variety. I've attached an image of the proof, and cannot seem understand ...
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131 views

Two questions on the Grothendieck ring of varieties

1) In the definition of the Grothendieck ring of varieties over a field $k$, which definition of the various notions of "variety" is chosen? Finite type and separated, or maybe more? 2) If ...
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31 views

Locally free sheaves on $\mathbb{A}^n$-bundles

Let $X$ be a variety, and let $p: E \rightarrow X$ be a $\mathbb{A}^n$-bundle. By this, I mean there is an open cover $U$ of $X$ such that if we base change to the open cover, $E \times_X U \simeq U ...
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1answer
43 views

Ring of regular functions on a point

Let $X \subset \mathbb{A}^n$ be an affine variety. Then the ring $\mathcal{O}_X$ of regular functions on $X$ is $A(X) := k[y_1,\dots,y_n] / I_X$, where $I_X$ is the vanishing ideal of $X$ (and $k$ is ...
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14 views

Question on cone of projective algebraic set $V$

Suppose I have a projective algebraic set $V \subseteq \mathbb{P}^n$ and its cone $C(V) \subseteq \mathbb{A}^n$. I was wondering about the following statement: $V$ is irreducible if and only if $C(V)$ ...
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1answer
48 views

Projections of the twisted cubic curve from points

I. Consider the twisted cubic curve $C$ in $\mathbb{P}^3$, given as the image of the veronese map $v_3: \mathbb{P}^1 \rightarrow \mathbb{P}^3$. Let $p \in \mathbb{P}^3 $ and consider the projection ...
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44 views

Is there a natural map $\Omega^1_X\to N^{\vee}_{Y/X}$?

Let $X$ be a smooth complex manifold and $Y\subset X$ be a complex submanifold. Is there some natural map from $\Omega^1_X$ to $N^{\vee}_{Y/X}$?
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1answer
22 views

Find order of elliptic curve

Given a prime $p$ such that $3$ does not divide $p-1$, what is the order of the elliptic curve over $\mathbb{F}_p$ given by $E(\mathbb{F}_p)=\{ (x,y) \in \mathbb{F}_p^2 | y^2=x^3+7 \}$ I thought if ...
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1answer
27 views

Existence of a Cech cover for computing Picard group

Let $X$ be a variety -- one can compute $\text{Pic}(X) = H^1(X, \mathcal{O}^*_X)$ by choosing a Cech cover which is acyclic with respect to $H^\bullet(-, \mathcal{O}^*)$. Can one always do this? It ...
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60 views

Valuative criteria with varieties

Let $X$ be an algebraic prevariety over an algebraically closed field $k$ (I.e. an integral scheme of finite type over $k$). In the valuative criteria for separatedness and properness of $X$ over ...
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80 views

What does Hartshorne mean here in Proposition 2.3?

I was re-reading Hartshorne, proposition 2.3 on page 73. He says: Now for any open set $V\subset \operatorname{Spec}A$ we obtain a homomorphism of rings ...
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1answer
50 views

Is the push-forward of a quasi-coherent sheaf under open immersion still quasi-coherent?

My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ ...
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86 views

Geometry Question with irregular hexagons

Suppose you have a rectangle with sides $x$ and $y$ and both numbers are integers and have no factors. now draw lines inside this rectangle starting with a line at $45^\circ$ coming out of a corner ...
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55 views

Proof of the formula for dimension of moduli of stable vector bundles on smooth curves

Let $C$ be a smooth curve of genus $g \ge 2$ over an algebraically closed field of positive characteristic. If I understand correctly, the dimension of the moduli space of vector bundles on $C$ of ...
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1answer
104 views

Lines which intersect the postive half axis of x

We have to find out which lines intersect the positive half axis of $x$. According to this formula we can determine if the angle between two points $(A[x_1, y_1]$ and $B[x_2, y_2]$ ) of the line ...
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2answers
94 views

When does the regularity of $A$ implies the regularity of $A[w]$?

Let $A$ be a commutative noetherian ring (I do not mind to assume that $A$ is a UFD), and assume that $A$ is regular. Recall that a commutative noetherian ring is called regular if all its ...
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73 views

A basic question about closed set in Zariski topology

Suppose I have homogeneous polynomials $f_1, .., f_r \in \mathbb{C}[x_1, ..., x_n]$, and let $I = (f_1, ..., f_r)$. Let $V:=V(I) \subseteq \mathbb{C}^n$ be the points where $f_i$'s vanish. Suppose $V$ ...
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1answer
35 views

Question about basic properties of degrees of algebraic sets

I am learning about degrees of algebraic sets at the moment, and in an article I am reading I came across the following: Let $V_i \subseteq \mathbb{C}^n$ be a hypersurface of degree at most $D$ for ...
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1answer
47 views

Finding the Krull dimension of a quotient ring $\mathbb{C}[x,y,z]/I$

I am interested in finding the Krull dimension of the quotient ring $A$ defined as follows: $$ A = \mathbb{C}[x,y,z] / (f_1, f_2, f_3), $$ where $$ f_1 = \frac12 y^3 z - (z-1) - xy $$ $$ f_2 = y^2 z^2 ...
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133 views

Why is the morphism induced by this linear system birational?

I have seen (what seems to be) the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a ...
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139 views

how much differential structure can we put on countable manifolds?

The motivation for this question is that I would like to formulate Lagrangian mechanics in a purely discrete setting (see also my older question at physics.se). Unfortunately several key pieces of ...
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34 views

On a sufficient condition for a closed morphism of schemes to be affine

Let $f \colon X \to Y$ be a closed morphism of schemes (i.e., the image of any closed subset of $X$ under $f$ is closed in $Y$). Let $y \in Y$. Consider the following assertions: (i) There is an ...
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1answer
38 views

Fulton, algebraic curves exercise 4.11

I'm doing the exercises of the book Fulton Algebraic curves and I'm stucked in the following problem: A subset $V\subset\mathbb{P}^n(k)$ is a linear subvariety of $\mathbb{P}^n(k)$ if ...
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1answer
507 views

Gaining insight into the Inverse Image Sheaf

Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is ...
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1answer
76 views

Any no-zero homomorphism of holomorphic vector bundles over a compact Riemann surface factors through a maximal rank homomorphism.

I was reading the paper "Stable and Unitary vector bundles on a compact surface" by Narashiman and Seshadri. I quote from the paper- Can someone please explain how does any non-zero homomorphism ...
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102 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
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1answer
106 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
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73 views

What is the convention for the codimension of an empty set?

I am learning about dimension and codimension of algebraic sets at the moment. I know that if $V \subseteq \mathbb{C}^n$ is an algebraic set defined by polynomials $f_1, ..., f_r \in \mathbb{C}[x_1, ...
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1answer
28 views

Equation of the locus

Find the equation of the locus of a point $P = (x, y)$ when the sum of the squares of the distances from $P$ to the points $(a, 0)$ and $(-a, 0)$ is $4b^2$, where $b \geq \dfrac{a}{\sqrt{2}}$?
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68 views

Equation of 27 lines on a cubic surface [closed]

For a smooth cubic surface $S$ in $\mathbf{P}^3$, there're always $27$ lines on it, with the same configuration. We know the automorphism group of the lines is not solvable. How do we show the ...
3
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1answer
60 views

Pushing forward vector bundles on a plane curve via projection from a point

Let $C \subset \mathbb{P}^2$ be a smooth plane curve, $P \in \mathbb{P}^2$ is point not on $C$, consider projection from this point $$ \pi :\mathbb{P}^2 - \{P\} \to \mathbb{P}^1, $$ and restrict this ...