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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Dual curve of an algebraic curve in affine coordinates

$F$ is an irreductible algebraic curve and we consider the application that sends a non-singular point $(a : b : c)$ in homogenius coordinates, to $\phi(a, b, c) = \bigl(f_x(a, b, c), f_y(a, b, c), ...
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9 views

Derivations on a plane curve with singularites.

I am looking for some general results on the Lie algebra of derivations $Der(R)$ of $R=\mathbb{C}[x,y]/(f)$. How to describe $Der(R)$ for $f=x^3-y^2$ or $f=xy$. Is it possible to characterise the ...
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1answer
58 views

Image of divisors under étale cover

Let $f\colon X\to Y$ be an étale cover of degree $d$ between two smooth projective varieties. If $V\subset X$ is an effective reduced and irreducible divisor, does $f$ restrict to an isomorphism $V\...
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48 views

A question on partial-derivative

Let $f(x,y) = \sqrt {{x^2} + {y^2}} p(x,y) + q(x,y)$, and $p$ and $q$ are two polynomials(non zero) Is it true that; $\frac{{\partial f}}{{\partial x}}$ and $\frac{{\partial f}}{{\partial y}}$ don't ...
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1answer
68 views

How can we prove, by Bézout's theorem, that $L$ has finitely many singularities?

Bézout's theorem Bézout's theorem for curves states that, in general, two algebraic curves of degrees $m$ and $n$ intersect in m·n points and cannot meet in more than $m·n$ points unless they have a ...
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75 views

Parametric Interpolation in the Plane

Given $i+j$ points in the plane, when can we find $x(t),y(t)$, polynomials of degree $i$ and $j$ respectively such that the parametric curve $(x(t),y(t))$ goes through each point? We can do this ...
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1answer
38 views

Order and residue of 1-form $x^{-1}dx$

Consider the rational 1-form $x^{−1}dx$ on $\mathbb{P}^1$. I am asked to compute its order and residue at all $P \in \mathbb{P}^1$. Could somebody help me with this? I do not really how to start ...
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1answer
34 views

The set of commutative matrices is an irreducible algebraic variety

Let $A, B$ matrices $n \times n$. Let $X = \left\{(A, B) \in \mathbb{A}^{2n^2} \mid AB = BA \right\}$. Prove that $X$ is algebraic and irreducible variety.
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56 views

Computing the Picard group of Number Fields in analogy with genus $0$ curves

It is possible to compute the Picard group of (some? all?) genus $0$ curves in the following manner: For concreteness let, $A = k[x,y]/(x^2+y^2-1)$ and $X = \operatorname{Spec} A$. Let $Y$ denote $\...
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1answer
74 views

Some elements of the function field of the Fermat curve

For $n>0$, consider the Fermat curve: $$C(n): \{X^n+Y^n=Z^n\}\subset\mathbb P^2(\mathbb C)$$ the function field of $\mathbb C(n)$ can be explicitly described in the following way. It is the set of ...
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2answers
105 views

Intersecting two pencils of plane curves

In $\Bbb{P}^2$, let $D_1,D_2$ be two curves of degree $d_1,d_2$ respectively. Choose two pencils $|D_1(t)|\subset|D_1|$ and $|D_2(t)|\subset|D_2|$ (free of fixed components) parametrized by the same ...
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2answers
60 views

Degree of a morphism from a curve to $\mathbb P^1_\mathbb C$: explicit description

Let $f:X\to \mathbb P^1_{\mathbb C}$ be a non-constant (i.e. surjective) morphism (of $\mathbb C$-varieties/schemes) from a smooth complex projective curve to the projective line. The degree of the ...
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1answer
60 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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45 views

Some Basic Properties of Tropical Amoebas

I am working through Sturmfels' new book on Tropical Geometry with another student, and we are stuck at a pretty important concept, namely the basic properties of amoebas. Let me reproduce a bit of ...
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18 views

Clarification in a theorem statement regarding intersection of Complex Algebraic Curves in $P_2$

I have a theorem in the book Complex Algebraic Curves- Frances Kirwan : *If two projective curves $C$ and $D$ of degrees $n$ and $m$ respectively in $P_2$ intersect at exactly $n^2$ points and if $...
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16 views

Non Vanishing Thetanulls

Let $C$ be a smooth curve over $\mathbb C$, a theta characteristic $\kappa$ is said to be non-vanishing if $h^0(\kappa)=0$. Does there exist always a non vanishing theta characteristic?
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19 views

If ${L_1} \subseteq {L_2}$ and and $L_1$ is union of continuous curves and $L_2$be algebraic curve. Can we say that $L_1$ is piecewise $C^∞$ curve?

Let $L_2=\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. If ${L_1} \subseteq {L_2}$ and $L_1$ is union of continuous curves. Can we say that $L_1$ is piecewise $C^∞$ ...
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1answer
36 views

Can we say that, every algebraic curve is a piecewise ${C^\infty }$ curve?

Let $L$ be algebraic curve. Can we say that, $L$ is a piecewise ${C^\infty }$ curve?
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23 views

Let $\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. Why does this algebraic curve, is continuous?

Let $\{(x,y):x,y\in R,f(x,y)=0\}$ be algebraic curve, and $f(x,y)$ is polynomial. Why does this algebraic curve, is continuous?
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42 views

Compute the genus of a curve with a flex point

The genus of a smooth plane curve is $g=\frac{(d-1)(d-2)}{2}$ and I know that if the curve has $n$ nodes the genus decreases by $n$. What happens if the curve has singular (non ordinary) points? In ...
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1answer
64 views

Euler characteristic singular surface

The setting is the one of algebraic curves over the complex numbers. It is known that in an irreducible nodal curve each node reduces the arithmetic genus by one: if $\tilde{C} \rightarrow C$ is the ...
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33 views

Projective Mapping of a Sphere

Show that the projective completion of the curve $Y=X^2$ is topologically a sphere. Consider the parametrization $X=t, Y=t^2$, where t ranges the sphere $\mathbb C\cup {\infty}$. How do I prove this ...
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1answer
50 views

Maps to $\mathbb{P}^1$ induced by rational functions.

I am reading from Hartshorne, Corollary II.6.10 page 138. Given a nonsingular (is this necessary?) curve $X$ over a field $k$, let $f\in K(X)^*\setminus k$. Then the inclusion of fields $k(f)\...
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19 views

A question on algebraic curve

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(z) = A_m z^m + \cdots + A_1 z + A_0$ is a matrix polynomial, and $z $ is a complex variable. $s_1 \ge s_2 \ge \cdots \ge s_n$ ...
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1answer
27 views

Image of morphism between curves

I have this projective curve $C\dots y^2z=x(x+2z)(x-z)$ and I have function $f$ on $C$, ie $f\in k(C)$ given by $f(x:y:z)=(y:z)$. What would be image of this function? Can I see it in some way as ...
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1answer
52 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
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89 views

Geometric interpretation of algebraic property

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, \...
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1answer
56 views

When is a hypersurface rationally connected?

A projective variety $X$ is said to be "rationally connected" if any two points on it can be connected by a map $\mathbb P^1 \to X$. Let $X$ be a smooth hypersurface in $\mathbb P^n_k$ defined by a ...
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54 views

Smooth completion of algebraic curves

I am having trouble understanding the concept of smooth completions of algebraic curves. I know the definition (smooth, complete curves which contain the curve X as an open subset ) and according to ...
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1answer
35 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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33 views

The set of curves of degree $d$ with two singular points or a degenerate singular point is closed?

Suppose $d>0$, $k$ is a field and $L_d$ is the space of projective plane curves over $k$ of degree $d$, namely $L_d=\mathbb P(k[X,Y,Z]_d)$, where $k[X,Y,Z]_d$ is the vector space of homogeneous ...
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37 views

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 G)...
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1answer
50 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 ...
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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 $F(\...
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1answer
44 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 $C$ ...
2
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1answer
109 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
71 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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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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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: K(...
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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
31 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
40 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
68 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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49 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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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 ...
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1answer
93 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 $z$: $(z)...
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39 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 = \mathbb{I}(\...
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1answer
72 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 ...