An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...

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Intersection multiplicity inequality problem

I have a question about an inequality that was stated in my class but never proven. We stated p is simple $I_p(F \cap G+H) \ge \min ( I_p (F \cap G) , I_p(f \cap H))$. Where we defined $I_p(F \cap ...
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1answer
46 views

Question about intersection multiplicity of a curve and it's tangent line

If we have a double point $a$ on some complex curve, call it $C$, defined by some polynomial $f$ and we have only one tangent line at $a$, call it $T_l$, then the intersection multiplicity $I(a,f \cap ...
3
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46 views

Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of ...
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1answer
38 views

Find the generator of the maximal ideal of regular functions on a curve on P

I'm stuck at a probably very simple exercise : Consider the algebraic curve $C: Y^2=X+X^3$ and $P=(0,0)$. Find a generator of the maximal ideal of the local ring of rational functions on $C$ ...
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1answer
108 views

How can find points such that the tangent fails to exist?

Let $F(x,y)=0$ and $F(x,y)$ be polynomial in $\mathbb R$. How can find points such that the tangent fails to exist ?
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1answer
63 views

The independence of the degree of morphisms between two curves on field extension.

Suppose $C_1$ and $C_2$ are two regular projective geometrically irreducible curves over $k$ and $F$ is a surjective morphism between them, then the degree of $F$ is the degree of the field extension ...
2
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1answer
35 views

Must a polynomial function of $x$ pass through infinitely many integer lattice points?

I made a mistake in my formulation of this question when I last asked it and got downvoted because the answer was actually trivial. However, I think the intended question is actually an interesting ...
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33 views

Does every line through the origin in the plane intersect the integer lattice an infinite number of times? [closed]

Question is in the title. What about every algebraic curve through the origin? Does every line through the origin in the complex numbers pass through an infinite number of Gaussian primes? EDIT: Just ...
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45 views

Pushforward on principal divisors

I am having much difficulty proving the following fact. Let $\phi:C_1\to C_2$ be a Galois cover of curves over an algebraically closed field $K$. Let $G=\mathrm{Gal}(K(C_1)/\phi^*K(C_2))$. Let $N: ...
3
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56 views

Automorphism of elliptic curve, Vakil 19.10.E

There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's ...
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1answer
28 views

A 3D curve correlation

Forgive me if its too basic, but i am looking to read some materials about a subject in which i don't know its name/field. So what we need to do, is to get a 3 axises curve, with unknown shape, that ...
2
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1answer
36 views

Automorphism group of genus 1 curve

Suppose $E$ is a regular curve of genus 1 over a field $k$ that is not necessarily algebraically closed. The automorphism groups of $E$ forms a one parameter family. If $k=\mathbb{C}$, then $E$ could ...
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1answer
58 views

Picture of elliptic curve double covers Rieman sphere

For simplicity, let us assume we live in the field $\mathbb{C}$. Then all elliptic curves are hyperelliptic, so they admit a double cover of $\mathbb{P}^1$ branched over four points, whose preimage ...
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45 views

Automorphism group of genus 2 curve

Suppose C is a genus 2 curve over a field k such that char k is not 2. Is there an easy way to show that the automorphism group is finite? If we assume k is algebraically closed then C is ...
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0answers
22 views

compute order x/z in Q

Let $x^3+y^2z+yz^2+\alpha^3z^3=0$ be a curve over $F_{16}$ where $\alpha$ is primitive element for $F_{16}$ and $\alpha^4+\alpha+1=0$. The point $Q=(0:1:0)$ is only infinity point this curve. I want ...
2
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1answer
79 views

Dimension of the Riemann-Roch Space of a divisor of a cubic (Fulton's Algebraic Curves Exercise 8.10)

Let $C$ be an irreducible projective cubic on $k[X,Y,Z]$, $k$ algebraically closed. Let $x = \frac{X}{Z}$, $y = \frac{Y}{Z}$, $z = x^{-1} = \frac{Z}{X}$. Also define the divisor of zeros of $z$: ...
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0answers
38 views

Definition of Numerical Trivial Invertible Sheaf

I am reading Ravi Vakil's book, Foundations of Algebraic Geometry and in section 18.4.9, the definition of numberically trivial invertible sheaf is, Suppose $X$ is a proper $k-$ variety, and ...
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47 views

Understanding Localized Rings mod an ideal.

Hi guys I am working with Fulton's book and I am trying to understand for myself the elements of two rings. $O_p(\mathbb{A}^n)/JO_p(\mathbb{A}^n)$ and $O_p(V)/\bar{J}O_p(V)$ Where $I = ...
1
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1answer
60 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
6
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1answer
148 views

Understanding an exercise from Fulton's Book on Algebraic Curves

I am reading Fulton's book Algebraic Curves. Currently I am working on a specific problem (2.43), and I have doubts about my work and would appreciate another opinion(s). Assume $p$ is the origin ...
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2answers
52 views

Why the image of quadratic Veronese map has the form $v\cdot v$?

It says that the image of quadratic Veronese map $v_2(P^1)$ is the subset of $P(Sym^2V)$ with the form $v\cdot v$. Isn't it has the form $x^2+xy+y^2$? So how can it be some $v\cdot v$?
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56 views

Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
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1answer
44 views

Irreducible quartic curve in projective space

Let $V$ be an irreducible quartic curve in $\mathbb P^2(\mathbb{C})$. Then $V$ cannot have a point of multiplicity greater than or equal to $4$. Furthermore if $V$ has a triple point $(a,b,c)$, ...
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62 views

Question about projective closure of a curve

I am working on a problem, and I wanted someone to look over my work and comment of I am on the right track. I have a polynomial $f\in K[x]$, where $K$ is an algebraically closed field, and $d=\deg ...
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1answer
67 views

Isomorphism of the affine circumference over certain fields.

Let us consider the coordinate ring of the circumference $$ A:=K[X,Y]/(X^{2}+Y^{2}-1), $$ and let us suppose that $K$ is infinite but not necesarilly algebraically closed. I wonder if $A$ is ...
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1answer
26 views

Ideal of the hyperbola in a field that is not algebraically closed.

Let $K$ a field not necessarily algebraically closed. I would like to find the coordinate ring of the hyperbola $$ V(XY-1)\subseteq K^{2}. $$ If the field was algebraically closed we could use ...
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37 views

Definition of algebraic cusp

Is not it true that for planar curves, an $\textit{algebraic cusp}$, say at the origin, is the one that can be locally represented by $y^n=x^m$ with $m,n\in\mathbb N$ and $(m,n)=1$? What is a ...
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25 views

Line through 2 flex passes through a third flex

(in $\mathbb P^2$)Show that a line through two flexes on a cubic passes through a third flex. I've tried to solve this problem using the corollaries of the Max Noether theorem (which talks about ...
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38 views

an exercise of shafarevich (I.1.8)

I have a question related to the following exercise of Shafarevich's Basic Algebraic Geometry: (I.1.8) Prove that for any nonsingular points $P_1,\ldots,P_r$ of an irreducible curve and numbers ...
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1answer
60 views

How to find the equation of the curve defining the intersection of two quadrics.

Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that ...
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32 views

A curve has the equation $y=x^3 - 4x^2 - 3x + 17$. What are the x-coordinates of the point(s) on this curve where the tangent is parallel to 4y=7x-11.

A curve has the equation $$y=x^3 - 4x^2 - 3x + 17$$. What are the x-coordinates of the point(s) on this curve where the tangent is... (a) parallel to $4y=7x-11$ (b) horizontal
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51 views

Does anyone know of any good sources on the algebraic theory of abelian varieties?

I have a copy of Mumford's book, but as a final year undergraduate I am finding it to be a little too dense as a starting text. Something lighter would be appreciated to get an intuition before ...
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18 views

Finding a cubic function with one real root given its graph.

When given a cubic graph with one real root. I need to find the equation of that graph using the function $$y=a(x-s)(x^2+bx+c)$$ where a, b,and c are unknowns. The y intercept is therefore $t = -asc$ ...
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0answers
36 views

Linearly normal embedding and varietes lying on quadrics

Let $X\subset\Bbb{P}^N$ be a smooth algebraic variety and assume that $X$ is not contained in a hyperplane. The embedding $i\colon X\hookrightarrow\Bbb{P}^N$ is called linearly normal if the linear ...
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23 views

Finding and analyizing the singularities in Affine and Projective space

Hi guys I am working this $F(x,y,z)=xy^4+yx^4+xz^4$ I need to find the singularities in affine and projective space and find the multiplicity of them.I would really appreciate some help tips. So ...
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1answer
20 views

Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got ...
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1answer
29 views

There exist cuspidal cubic sections in a nonsingular cubic surface in $\mathbb{P}^3$

This is part of Exercise 7.3 in Undergraduate Algebraic Geometry by Reid. Let $S: (f=0) \subset \mathbb{P}^3$ be a nonsingular cubic surface. For $P\in S$ prove that if $P$ is not on a line of $S$ ...
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1answer
29 views

Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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1answer
12 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...
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39 views

Integers characterizing singularities of algebraic curves

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field ...
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1answer
35 views

Proving a projective quadric is nonsingular

Let $K$ be an algebraically closed field of characteristic $\neq 2$. Let $C$ be an irreducible quadric curve in $\mathbb{P}^2$, i.e. $C = Z(F)$ where $F$ is an irreducible degree 2 form. I think we ...
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1answer
64 views

Finding length of curve $y^2 = 64(x+3)^3$ for $0 \le x \le 3$

Not getting the right answer for this, can someone point me to where I'm going wrong?
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40 views

Abstract regular curve over non-algebraically closed field

In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves ...
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1answer
77 views

Image of a line or conic on Veronese surface.

This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid: Consider the Veronese surface $S$ defined by the map: $$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$ where ...
2
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1answer
44 views

Affine variety over a field which is not algebraically closed

I am now trying to prove the following statement. If the field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}$ can be written as the zero set of a single polynomial in ...
2
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1answer
64 views

Is the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ birational?

This is Exercise 5.3 (a) in Undergraduate Algebraic Geometry by Reid. Does the map $\phi: \mathbb{P^2\rightarrow \mathbb{P^1}}$ defined by $\phi(x,y,z)=(x,y)$ define a rational map? Determine ...
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32 views

Does this base change yield another dominant morphism?

Here's something that seems to be true, or at least I hope it to be true, but I'm unable to prove it: Let $S$ be a $k$-rational surface and $B$ a curve, both projective, smooth and geometrically ...
3
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71 views

Question on a “dual form” of $F$ that caprures the singularity of a plane section of $F$

In an article I was reading the notion of dual form came up, which I write down below. I was interested in learning more about this and I have two questions regarding it. This is how it came up: ...
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1answer
40 views

Irreducibility of an affine variety in an affince space vs in a projective space.

Proposition 5.5 in Undergraduate Algebraic Geometry by Reid says (I only write down a brief idea since the proposition is long and involves some other notations to define): The affine variety $U$ ...
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36 views

$f\in k(\mathbb{A^2})$ not regular at the origin implies it is not regular at points of a curve passing through the origin.

This is Exercise 4.12 (a) in Undergraduate Algebraic Geometry by Reid. Prove that any $f \in k(\mathbb{A}^2)$ which is not regular at the origin $(0, 0)$ also fails to be regular at points of a ...