The study of geometric objects defined by polynomial equations. Algebraic curves, such as elliptic curves, and more generally algebraic varieties, schemes, etc.

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Resolution of Singularities: Base Point

Consider the curve $y^2=4x^3-ax-b$, where $a$ is a fixed constant and $b$ is a free constant. For each value of $b$ we get a family of curves. Part 1: Show that the family of curves intersect at ...
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52 views

Problem I.3.18 in Hartshorne

Problem I.3.18b-c in Hartshorne is concerned with the surface $Y$ of $\mathbb{P}^3$ given parametrically by $(x,y,z,w) = (t^4,t^3u,tu^3,u^4)$. In particular, part c asks to prove that $Y$ is ...
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43 views

Nakayama lemma and intersection multiplicity.

We have the theorem that $I_{p}(f,g)\ge m_{p}(f)m_{p}(g)$ with equality iff $f$ and $g$ have no tangent lines in common. 1) Using Nakayama's lemma prove this in the case that $m_{p}(f)=1$. 2) ...
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1answer
43 views

The nonexistence of a polynomial

I'm studying algebraic geometry. To illustrate a nonalgebraic set, it is given that a unit circle except for a point on it in cartesian product or whole plane except for one point. Why doesn't a ...
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3answers
100 views

How to imagine “tensoring with Serre's twisted sheaf”

What has an algebraic geometer in mind when (s)he sees $\otimes \mathcal{O}(1)$? I think it has something to do with an intersection of a hypersurface...? Thanks, Adrian
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53 views

Finitely many singular points of an irreducible polynomial

let $k$ be a field, and consider an irreducible polynomial $f∈k[x,y]$. Let $S(f)$ denote the singular points of $f$ (points that are simultaneously zero on $f$, the $x$-derivative of $f$, and the ...
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58 views

“Implicit representations” of algebraic varieties

Consider a system of polynomial equations $S$ in multiple variables $x_1,\dots,x_n$ over the field $\mathbb{C}$. Is there a simple characterization of when the following property holds: There exists ...
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1answer
62 views

Veronese surface contains no lines

Why does Veronese surface contain no lines? Can you give me a reference about this fact? Thank you for your answers.
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1answer
142 views

Proof of rigidity lemma

I am trying to understand in full details the proof of rigidity lemma as proved here http://staff.science.uva.nl/~bmoonen/boek/DefBasEx.pdf [Lemma 1.11, Pag. 12]. The statement in this reference is ...
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38 views

The general expression of plane through the intersection of other two planes

For two planes: $$A_{1}x+B_{1}y+C_{1}z+D_{1}=0 $$ $$A_{2}x+B_{2}y+C_{2}z+D_{2}=0$$ Prove that any plane going through the intersection line of the previous planes could be expressed like where ...
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32 views

Finding the point satisfying the condition

Given N interesting points on the plane. Each interesting point has integer coordinates. Also, all the interesting points form a strictly convex polygon. If we select two coordinates from these ...
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35 views

Question regarding function field

I have learned in my algebraic curves class that the function field is the field of rational functions on a curve $C$ (or some variety). I was at a number theory talk, where the person counted the ...
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1answer
834 views

How to find any exterior angle of an irregular polygon formula?

So I took a challenge from my Geometry teacher to create code that when the user gives the computer how many angles / sides a polygon has and the angle of each of the interior angles it could find ...
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1answer
71 views

Number of points on an elliptic curve over $ \mathbb{F}_{q} $.

I have the following elliptic curve: $$ E: \quad Y^{2} = X^{3} + 1 ~ \text{over} ~ \mathbb{F}_{q}, ~ \text{where} ~ q \equiv 1 ~ (\text{mod} ~ 3). $$ I want to know the number of points on this curve. ...
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1answer
46 views

Standard proof that the set of sigularities is closed

I am trying to prove that the set of singular points of an affine variety is closed. Suppose $X\subset \mathbb{A}^n$. As every affine tangent space $T_x$ is embedded in this $\mathbb{A}^n$ we consider ...
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21 views

Some elementary questions on biprojective spaces

Suppose we define projective spaces over some field $k$, and consider the product $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2}$. Unlike the affine case, we have $\mathbb{P}^{n_1} \times \mathbb{P}^{n_2} ...
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1answer
35 views

Describing a tangent cone. What is that?

Could you please explain what a tangent cone is? For instance, consider the curve on $\mathbb{A}^2$ given by $f(x,y)=x^2-y^3=0$. Linear part is zero cause $\frac{\partial f(0)}{\partial ...
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2answers
109 views

When do we have $\operatorname{depth}_{A} B = \operatorname{depth}_B B$?

Let $(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism with $A$ a regular local ring. Assume further that this ring map is finite. How can we prove that $\operatorname{depth}_B B = ...
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1answer
68 views

Question about Qing Liu's Algebraic Geometry book

I was just wondering what the real prerequisites are for reading Qing Liu's 'Algebraic Geometry and Arithmetic Curves', and if it is a good first book on the subject. In his preface he states that the ...
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33 views

Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
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1answer
40 views

Tangent bundle for the projective plane curve

Consider the cubic $C$ with an equation $x_0^3+x_1^3+x_2^3=0$ (this is a projective curve on $\mathbb{P}_2=\mathbb{P}(V)$). I need to find the equation of the closure of all tangents to $C$ (it is ...
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2answers
47 views

Question regarding morphism of ringed spaces

I have recently started studying schemes, and I have encountered this passage from the book by Kenji Ueno: My questions: i) If $(X,O_X)$ is a local ringed space, why is $(X, i_*({O_X}_{|U}))$ also ...
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1k views

Algebraic geometry project ideas for high school students

I am teaching a "senior seminar" course for strong students at our local high school. For 6 weeks the students learned about basic/classical algebraic geometry. In a few weeks they will start projects ...
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18 views

Rank of derivative polynomial map equals dimension image?

I've been told that given a polynomial map $f:X\to Y$ in characteristic zero, there exists an open dense subset $U$ of $X$ such that for all points $x$ in $U$, the rank of the derivative of $f$ in $x$ ...
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64 views

Prove: $U \mapsto \mathrm{Hom}(U, Y)$

Rewording this problem via what Zhen Lin's notion of the original question is. For $X$ and $Y$ ringed spaces Prove: For each open $U \subset X$ the Presheaf $U \mapsto \mathrm{Hom}(U, Y)$ is a ...
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1answer
58 views

3 points collide in $\mathbb{C}^2$

In Nakajima's book, "Lectures on Hilbert Schemes of Points on Surfaces", he gives an explicit description of the corresponding ideal for two points colliding in $\mathbb{C}^2$. This basically ...
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1answer
59 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
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1answer
49 views

What is a Presheaf (intuitively) and help with the technical machinery.

I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if ...
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0answers
37 views

Variety $V=\{(x,y)\in k^2\mid xy=1\}$ is connected

Let $k$ be an infinite perfect field. Show that the variety $$V=\{(x,y)\in k^2\mid xy=1\}$$ is connected of dimension $1$. Many thanks in advance.
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1answer
38 views

Degree of ample bundle over projective curve is positive

(From Vakil's notes, Exercise 18.4.K) If $C$ is an integral projective curve over a field $k$, and $\mathscr{L}$ is an ample line bundle on $C$, why is the degree of $\mathscr{L}>0$? If $C$ is ...
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38 views

Unstable extension rank two vector bundle

I'd like to classify the not semistable extensions of this form: $$0 \to O_C \xrightarrow{i} E \xrightarrow{\pi} L \to 0$$ where $C$ is a curve and $L$ is a line bundle isomorphic to the determinant ...
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2answers
60 views

Equation $1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$

How to prove that the following equation: $$1+x^8y^4+x^4y^8-x^2y^4-x^6y^6-x^4y^2=0$$ has for solution(in real numbers): $|x|=|y|=1~$ only. Any hint would be appreciated.
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1answer
38 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
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1answer
32 views

What is the intuition behind contours and their geometric properties

What is the the intuition behind contours? Can someone explain whar are contours, their geometric properties in simple manner
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1answer
82 views

Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
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1answer
47 views

tensor product of presheaves of modules

Let $\mathscr{O}$ be a presheaf of rings on $X$ and $\mathscr{F}$, $\mathscr{G}$ be presheaves of $\mathscr{O}$-modules on $X$. Let $\mathscr{O}^{\#}$,$\mathscr{F}^{\#}$ and $\mathscr{G}^{\#}$ be ...
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77 views

Ex 6.7 in Görtz and Wedhorn's AGI

From Algebraic Geometry I, by Ulrich Görtz and Torsten Wedhon, p.165: Exercise $\boldsymbol{6.7}$. Let $k$ be a field, let $X$ be a $k$-scheme, and let $Y_1$ and $Y_2$ be closed subschemes of of ...
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1answer
54 views

cotangent space

Let $K$ be a field and $L$ be its field extension, $K\subset L.$ Let $V \subset L^n$ and let $\mathcal{I}(V)\subset K[x_{1},...,x_{n}]$ be its vanishing ideal. Let ...
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1answer
47 views

a question on higher direct images in a product

I want to compute $H^i(X,\pi_1^* O(a)\otimes \pi_2^*O(b))$ where $X=\mathbb{P}^1\times \mathbb{P}^1$ and $\pi_i$ is the projection onto the $i$-th $\mathbb{P}^1$ factor. I realize this is a special ...
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1answer
77 views

Ringed space: interpretation of quotient.

Let $(X,O_X)$ a locally ringed space, I'd like to define the tangent space $T_x$ for $x \in X$. We can consider the local ring (stalk) $R_x$ in $x$ with maximal ideal $m_x$ and from invertibility we ...
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1answer
46 views

Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
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1answer
94 views

Hartshorne “Algebraic Geometry” theorem 8.15

The theorem $8.15$(p.177) from the Hartshorne's book "Algebraic Geometry" says: " Let $X$ be an irreducible separated scheme of finite type over an algebraically closed field $k$. Then ...
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2answers
76 views

sections of an invertible sheaf, and their support

Suppose I have a section $s$ of an invertible sheaf $L$, vanishing along a divisor $D$. Then there is an isomorphism $(L, s) \simeq (O(D), 1)$. In the next paragraph I'll pick $D=K_X$, but that is ...
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2answers
91 views

Effective Cartier divisors

A Cartier divisor on a scheme $X$ is effective if it can be represented by $\{(U_i,f_i)\}$ where $U_i$ covers $X$ and $f_i \in X$ Let $\mathcal{I}$ be a sheaf of ideals which is locally generated by ...
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1answer
80 views

pullback of canonical divisor

Let $Y$ be a smooth variety of dimension $n$. Then I can get (a representative for) the canonical divisor class $K_Y$ on $Y$ by taking any rational $n$-form $\omega$ on $Y$ and taking its divisor of ...
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1answer
55 views

About globally generated Sheaves

On Vakil's Lecture Notes, he puts an important exercise that says: ''Suppose $\cal{F}$ is a finite type quasicoherent sheaf on a scheme $X$. Show that$\cal{F}$ is globally generated at $p$ if and ...
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23 views

The closure of semialgebraic sets is semialgebraic.

I want to prove that the closure of semialgebraic subsets of $\mathbb{R}^n$ with respect to the Euclidean topology is semialgebraic. I may use the Tarski–Seidenberg theorem. Please give me not the ...
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1answer
138 views

Hartshorne ex III.10.2 on smooth morphisms

I need some help with the following exercise: Let $f:X\rightarrow Y$ be a flat proper morphism between varieties over $k$, where variety means separated, finite type, integral, and $k$ not ...
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40 views

global section of some sheaves

Let $\mathrm{Grass}(r,V)$ be the Grassmannian over a field $k$. What is $H^0(\Sigma^{\alpha}(S))$, where $S$ is the tautological sheaf and $\Sigma$ the Schur functor. In characteristic zero this is ...
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Definition of quasi-projective variety and some related questions

I'm a little bit confused by Definition 1.64 on page 32 in this book. This definition says: A prevariety is called quasi-projective variety if it is isomorphic to an open subvariety of a projective ...