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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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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17 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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0answers
6 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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0answers
24 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
15 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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0answers
22 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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17 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 ...
2
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0answers
48 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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0answers
13 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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1answer
53 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 ...
4
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1answer
31 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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44 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
19 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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1answer
24 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 ...
2
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0answers
25 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
38 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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0answers
13 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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0answers
10 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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0answers
38 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
17 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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0answers
33 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
22 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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2answers
393 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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0answers
75 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 ...
5
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2answers
108 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
45 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
68 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
41 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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0answers
33 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
31 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
25 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 $b \geq \dfrac{a}{\sqrt{2}}$?
3
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0answers
63 views

Equation of 27 lines on a cubic surface [on hold]

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 ...
6
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0answers
92 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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2answers
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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0answers
45 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 ...
2
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2answers
84 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
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0answers
23 views

What is the formalism for a map that returns the adjacent vertex positions of a given adjacency matrix?

How do I formally denote a map that returns the adjacent vertex positions of a given adjacency matrix? Example: V = undirected adjacency matrix $$ V = \left[ \begin{matrix} v_{1,1} & v_{1,2} ...
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1answer
32 views

Fourier-Mukai kernels of mutations?

if I have an exceptional object E (on say the derived category of a smooth and projective variety) then I can define the left and right mutation functors. These are typically defined in terms of ...
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0answers
54 views

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curves second edition (J. H. Silverman) [closed]

Please I would like to have the solution of the exercise 3.7 on page 105 of The Arithmetic of elliptic curve second edition (J. H. Silverman)
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0answers
51 views

How to obtain the genus of the Riemann surface corresponding to an algebraic curve

I am trying to read about the genus of an algebraic curve. I have been told that there is a connection between topological genus and genus defined for an algebraic curve. Since an algebraic curve ...
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1answer
85 views

Basic question related to dimension of intersection of two varities

Let $V$ and $W$ be irreducible varieties in $\mathbb{C}^n$. I have learned that intersection $V \cap W$ satisfies the following: $$ codim \ V + codim \ W \geq codim \ V \cap W. $$ I was wondering if ...
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93 views
+200

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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1answer
97 views

Trying to use the Zariski topology in a problem without knowing scheme theory.

I don't know scheme theory, and I am doing a problem and the solution involves making conclusions based on the Zariski topology, and I want to make sure that I am "intuiting" things correctly when ...
2
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0answers
26 views

Basic question about the properties of dimension of an algebraic set

I am learning about the dimension of an algebraic set and I have a couple of questions I am hoping to resolve to have a better understanding. Let $V$ be an algebraic set in $\mathbb{C}^n$, defined ...
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0answers
37 views

How to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to$\sum_{P\in X} i_P(I_P)$

This a Hartshorne exercise (Ex II 1.21d) Let $X=\mathbb{P}^1$. Let $\mathcal{K}$ be the constant sheaf of the quotient field of X. Then we need to show that $\mathcal{K}/\mathcal{O}$ is isomorphic to ...
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36 views

Show that exist point $w \in \mathbb{C}$ with $w^2 \in K$ and $z \in K(w)$

Let K is subfield of $\mathbb{C}$ with $K=\overline{K}$. $F(k)$ is a set with all circles in complex plane with midpoint in K and radi^us equal distance between two points from K. Let $z\in ...
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0answers
16 views

Group action on closed subschemes

Let $G$ be a connected, linear, semi-simple algebraic group and $P \subset G$ the maximal parabolic subgroup. We know that $Z=G/P$ is a projective variety. Then, 1) Does $Z$ contain a line? 2) In ...
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0answers
40 views

Existence of Shafarevich maps(theorem 3.6) on Kollar 's book

I have some problem when reading Theorem 3.6 of Kollar's book Shafarevich Maps and Automorphic Forms, page 41 (Corollary 3.5 of this article ), which states that Let $X$ be a normal variety, ...
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1answer
105 views
+50

Counting points on the Klein quartic

In Moreno's book "Algebraic Curves over Finite Fields", he mentions the following in passing with no further comments ($K$ denotes the Klein quartic defined by $X^3 Y + Y^3 Z + Z^3 X = 0$): The ...
5
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1answer
56 views

Is simply connectedness preserved after deleting a high codimension set

Suppose $X$ is a complex manifold of complex dimension $n$, $Z$ is a subvariety of complex codimension at least $2$. Suppose $\pi_1(X)=0$, do we have $\pi_1(X-Z)=0$? Do we have $\pi_1(X-Z)=\pi_1(X)$ ...