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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linear systems and maps

Given a regular map $\varphi:C\to \mathbb P^n,P\mapsto \mathbb (f_0(P):f_1(P):\ldots:f_n(P))$, we can associate a linear system $|\varphi|$ in the following manner: let the divisor $D=-\min div(f_i)$ ...
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1answer
71 views

Example of a curve with this property

I'm reading Fulton's book and he defines the linear series $g_n^r$: So a curve $C$ is trigonal if it has a divisor which has a linear system $g_3^1$. I'm looking for a simple example of a trigonal ...
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145 views

Cancellation problem: $R\not\cong S$ but $R[t]\cong S[t]$ (Danielewski surfaces)

I would like to understand why the two rings $$ R={\mathbb{C}[x,y,z]}/{(xy - (1 - z^2))} \\ S=\mathbb{C}[x,y,z]/{(x^2y - (1 - z^2))} $$ are not isomorphic, but $R[t]\cong S[t]$. This example is ...
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46 views

Minimal free resolution of the twisted cubic

This is exercise 13.15 in Harris' book "A First Course...". Let $X$ be the twisted cubic with ideal $I(X) = (XZ-Y^2,YW-Z^2,XZ-YW).$ Let $S(X)$ denote the homogeneous coordinate ring of $X$ and ...
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37 views

For what functions is this theorem correct?

Theorem$_0$: If $g:\mathbb{C}^k \to \mathbb{C}^k$ sends $(t a_1,...,t a_k)$ to $(t^{\alpha_1}b_1,...,t^{\alpha_k}b_k)$ for all $t\in \mathbb{C}$, the preimage of any point has size $\alpha_1 \cdots ...
3
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78 views

Fiber dimension theorem for locally closed sets

I want to prove (or to find a reference to) the following statement: Statement: Let $Z$ be an irreducible locally closed set (Zariski topology) of $\mathbb C^n$ and $\pi$ be a projection on the ...
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122 views

A “generalized field” with $q$ elements, when $q$ is any number?

It is well-known that if a finite field has $q \in \mathbb{N}$ elements, then $q$ is prime power and $q > 1$. However, various modification of the concept of a "field" have been made in order to ...
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83 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
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1answer
128 views

Proof verification of a weak version of Bezout's Theorem

I'd like to make sure here that my reasoning seems sound. I am working from Kirwan's book on algebraic curves. I was not totally happy with her proof of this theorem, so I wanted to see if I could ...
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1answer
41 views

Cohomology of Severi-Brauer varieties

What can be said about Galois-module structure of $l$-adic cohomology of a Severi-Brauer variety over a local field? In particular, I'm interested in the proof of the proposition given at the top of ...
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32 views

Normal cone and specialization

This question is from the Kashiwara and Schapira's book: Sheaves on Manifolds Let M be a closed submanifold of X and let S be a locally closed subset of X, prove that C$_M$(S) = ...
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31 views

Projective variety defined by a non-radical ideal.

In the context of the Exercise 5.3.D in Vakil's notes, I want to show that there are examples of a reduced graded ring $A$ and a non-radical homogeneous ideal $I$ such that $\text{Proj}(A/I)$ is a ...
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0answers
29 views

A question about the pull-back and proper pushforward functors (convolution product of perverse sheaves) on $SL(2,\mathbb C)$

Let $G = SL(2,\mathbb C)$, which is an algebraic group of type $A_1$ over $\mathbb C$. Let $B$ be a Borel subgroup of $G$. Let $X =G/B$. Then $X \cong \mathbb P^1$ and it has a stratification $X = ...
3
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1answer
77 views

AG on non-Noetherian rings

I must apologize beforehand as this question is pretty basic, but I can't seem to find a satisfying answer in the introduction section of the book I'm currently reading (if there is a page on here, I ...
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1answer
44 views

Normal bundle of a section of a $\mathbb{P}^1$-bundle

Let $X$ be a normal projective variety over $\mathbb{C}$ and let $\mathcal{L}$ be an ample line bundle on $X$. If we define $P=\mathbb{P}_X(\mathcal{O}_X\oplus \mathcal{L})$, then the quotient ...
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1answer
91 views

What shall I write for a reason for applying graduate school for algebraic geometry?

I'm a undergraduate applying a graduate school this year and now I'm writing a letter of self-introduction. To be honest, I don't know what exactly is algebraic geometry and I think 99% of ...
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21 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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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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45 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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1answer
38 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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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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32 views

Show that the number of points of $V(I)$ is at most $m_1m_2…m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle ...
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42 views

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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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 ...
3
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1answer
70 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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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 ...
2
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1answer
43 views

How to reduce cubics in the plane to a canonical form?

I watched a video from Wildberger in the Differential Geometry series ( first, or third lecture, I don't remember ) where he says the following. The general format of a cubic curve is $$a x^3 + b ...
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1answer
39 views

Intersection product of line bundles with $\mathcal{F}$.

I am having trouble with computing the intersection number using the following definition [Ref Vakil Chapter 20]: Suppose $\mathcal{F}$ is a coherent sheaf on $X$ with proper support of ...
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172 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 ...
3
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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 ...
14
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114 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
42 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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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
55 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 ...
1
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1answer
34 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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32 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
44 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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16 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
58 views

Reference on Malcev completion

I need a reference for learning Malcev completion, its associated group scheme, and Lie algebra. Thanks!
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45 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
24 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 ...
10
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1answer
96 views

Is the group-theoretic Grothendieck-Springer resolution Calabi-Yau?

Any cotangent bundle is Calabi-Yau (by which I mean the canonical bundle is trivial), so the Springer resolution $T^*(G/B)$ is Calabi-Yau. I think that the Grothendieck-Springer resolution ...
4
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1answer
28 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 ...
10
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2answers
454 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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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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2answers
133 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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1answer
50 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 ...
3
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0answers
74 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
12 views

Bijection between dominant rational maps and morphisms of function fields?

Let $X$ and $Y$ be two integral schemes of finite type over a field $k$. Consider the function fields $K(X)$ and $K(Y)$. Do we have a bijection between: (a) Dominant rational maps $X \rightarrow Y$ ...