Tagged Questions

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When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
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Mobius transformation of Algebraic curve

I am working on the uniformization of algebraic curve problem. Currently, my adviser gave me a question about build a Mobius transformation between algebraic curves, and then lift it to the Rimeann ...
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Finding the area enclosed by $px^4+qxy+ry^2+sy+t=0$

When $px^4+qxy+ry^2+sy+t=0\ (p,q,r,s,t\in\mathbb R)$ represents a simple closed curve on the $xy$ plane, can we represent the area enclosed by this curve by $p,q,r,s,t$? If yes, then how? Example 1 ...
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Parabola tangent to four lines

Suppose that in the affine plane R^2 four lines are given, with the property that no two are parallel and no three are concurrent. Show that there exists a unique parabola tangent to each of the four ...
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Why is it called “elliptic” curve?

One of my favourite and most studied algebraic curve is the elliptic curve. But something that I have never asked myself is: Why do they call this nonsingular cubic curve an "elliptic" curve? ...
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Why a cubic plane curve meets a line three times?

Can someone explain to me why a cubic curve in a projective plane always meets a line three times?
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How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
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Birrational curves and singularities [closed]

If $C$ and $D$ are two birrational plane curves. Is there some relation between their singular points?
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Holomorphic Differentials on a non-singular curve.

So I've been working on this for an exam I have coming up and I'm not sure I really understand. If I have a curve defined by some homogenous polynomial P, I can show that the canonical divisor class ...
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Finiteness of morphism of curves with fixed image

This question comes from the proof of "bend and break" lemma in "Higer-dimensional algebraic geometry" (p.59-60). I use the notations in compatible with the notation given there for convenience. Let ...
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Are $k$ points on a smooth algebraic plane curve ever in general position?

Let $C$ be a smooth plane curve of degree $d$ and genus $g=\frac{(d-1)(d-2)}{2}$. Let us choose $k\leq g+3d-1$ points on $C$. Is it true that the dimension of the space of plane curves of degree $d$ ...
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An question on effective divisor (Clifford 'S theorem)

For an effective divisor $D\ge 0$ on a curve $Y$, define $$\lvert D\rvert =\{ D' \in \mathrm{Div}(Y) \mid D'\ge 0 \;\text{ and }\; D' \sim D \}$$ where $D\sim D$ means $\exists$ a rational ...
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A question on the morphism of projective varieties

The continuation of this, my question I want to show that $X$ and $Y$ are smooth and irreducible curves then $f(X)$ is either $Y$ or a point. Note that I know the proof of this ...
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the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m$

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m$is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. ...
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Can a “negative degree” line bundle on a reducible curve have global section?

Suppose $A,B$ are curves on smooth projective surface, having no common components and intersect, so $(A.B)>0$,do we have $H^0(O_A(-B|_A))=0?$ (here $A,B$ are effective divisors, may be irreduced ...
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intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
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local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
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Cubic curve in projective space

Is it true that every cubic curve in $\mathbb{P}^3$, which is not contained in a plane, can be parametrized by polynomials? $\\\\\\\\$
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How to decide if an ideal in $\mathbf Q[X,Y]/(P)$ is principal?

Let $P(X,Y)$ be an irreducible polynomial in $\mathbf Q[X,Y]$. Given an ideal $I$ of the quotient ring $\mathbf Q[X,Y]/(P)$ (say given by a set of generators) how can I decide if $I$ is principal or ...
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Automorphisms of rational curves

Let $X$ be a non-empty open subscheme of $\mathbb P^1_{\mathbb C}$. What is the automorphism group of $X$ in terms of PGL$_n(\mathbb C)$ and the points on the boundary?
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covering of projective curve by affine parts

For $\mathbb{P}^n$ we can let $U_i = \{(x_1:\cdots:x_i:\cdots:x_{n+1}) : x_i \neq 0\}$. Then let $C \subset \mathbb{P}^n$ be a projective plane curve. We can decompose $C$ into a union of affine plane ...
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Eliminating a parameter when intersecting a manifold with a hyperplane

In the Euclidean space $\mathbb R^4$ we look at the intersection of the equations$$x^2 + y^2 = 1 \\ z^2 + w^2 = 1$$ sometimes known as the Clifford torus. This is known to be a 2-dimensional manifold, ...
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I've been trying to understand the topological "link" between algebraic varieties and their associated knots/links and to this end I've been reading F. Kirwan's book, "Complex algebraic Curves". The ...
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definition of affine plane curve

Let $k$ be a field. On page 5 of Milne's Elliptic Curves, the author defines an algebraic curves to be defined by polynomials $f \in k[x,y]$ with no repeated irreducible factors in ...
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multiplicity of intersection of projective curves

This example is taken from Silverman and Tate's Rational Points on Elliptic Curves. Suppose we have the line $C_1: x+y=2$ and the circle $C_2: x^2 + y^2 = 2$. We see that these intersect at the ...
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Surjective morphism of complete non-singular curves is normalization

My syllabus on algebraic geometry states the following: ''Let $\phi: X \to Y$ be a surjective morphism of complete non-singular curves. Then $X$ is the normalization of $Y$ in the function field of ...
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Let $\Lambda=<2\omega_1,2\omega_2>$ be a lattice in $\mathbb{C}$, $C=\mathbb{C}/\Lambda$ an elliptic curve. Let $O(0,0)$ (neutral element of the lattice), and $A \in \mathbb{C}/\Lambda$ point of ...
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branched covering factors through a primitive one

I'm struggling with an assertion I found in an article I'm reading. A projective complex curve $X$ is rationally uniformized by radicals if there exists a branched covering $X\to \mathbb{P}^1$ such ...
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Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$
Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ ...