Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [algebraic-curves]

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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

If $K$ is a field, then an algebraic curve is an equation $f(X,Y)=0$ where $f(X,Y)$ is a polynomial with coefficients in the field. In other words, it is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

2500 questions
7
votes
2 answers

If $f$ has a pole, does $f^2$ has a pole?

I don't understand something in the exercise 2.17 of Algebraic Curves of Fulton. Let $k = \overline{k}$ a field and $V$ be the variety defined by the zero of $ I = ( y^2 - x^2(x-1) ) \subset k[x,y]$. Let $\overline{x}, \overline{y}$ be the…
user21319132
  • 71
4
votes
1 answer

inconsistency of the Plücker's formula

I'm a beginner in algebraic curves and as an exercise I'm playing with the Plücker's formula. I'm finding some inconsistency in these formulas and I would like to know where I'm wrong. We know the dual curve of a dual curve of $F$ is the $F$ itself.…
user122342
  • 355
4
votes
1 answer

The intersection numbers in Fermat curve

I'm a beginner in this subject and I think this "easy" exercise could help me to have more practice in basic algebraic curves. Let $F=X^{p+1}+Y^{p+1}+Z^{p+1}$ be a Fermat curve in the field $k$, with $char(k)=p\gt 0$. I've already showed every point…
user122342
  • 355
4
votes
1 answer

Function that is identically zero

Is it true that: Any rational function $f$ on $\mathbb{C}^2$ that vanishes on $S=\{(x,y)\in\mathbb{C}^2 : x=ny \text{ for some } n \in \mathbb{Z}\}$ must be identically zero. I have a theorem that says any rational function that vanishes on an open…
user44322
  • 993
4
votes
2 answers

Is the Oval ( based on Ptolemy inequality) known?

It has a property of enclosing quadrilaterals so the ratio of their diagonals product and sum of their opposite sides pair products is constant $(e<1)$. The curve is from a family defined by the Ptolemy Inequality In order to rope in the Ptolemy…
Narasimham
  • 40,495
4
votes
1 answer

Exercise 2.19 algebraic curves by william fulton

Let $f$ be a rational function on a variety $V$. Let $U = \{P\in V; f \textrm{ is defined at }P\}$. Then $f$ defines a function from $U$ to $k$. Show that this function determines $f$ uniquely. So a rational function may be considered as a type of…
Bruno Rodrigues dos Santos
  • 351
4
votes
1 answer

Finding a curve that intersects with $V(X_{0} X_{1}^{3} + X_{1}^{4} − X_{2}^{4})$ under certain conditions.

Let $D=V(X_{0} X_{1}^{3} + X_{1}^{4} − X_{2}^{4})\subset\mathbb{P}_{\mathbb{C}}^{2}$ and $C=V(X_{0} X_{1}^{2} + X_{1}^{3} − X_{2}^{3})\subset\mathbb{P}_{\mathbb{C}}^{2}$. I have got that $C\cap D=\{(0:1:1), (1:-1:0), (1:0:0)\}$ with…
Srinivasa Granujan
  • 1,270
3
votes
3 answers

Proof that two simultaneous line equations do not intersect?

Apologies if this isn't at the level of questions expected here! I've got two simultaneous equations to solve. (Equation 1): $ x y = 4 $ (Equation 2): $ x + y = 2 $ They produce the following curves: Question: Whilst it's graphically obvious that…
yellow-saint
  • 133
3
votes
1 answer

Why there is a minimal element of this set

I'm trying to understand this proof: I know intuitively, but Why formally there is such a minimal element? I need help Thanks
user42912
  • 23,582
3
votes
0 answers

Parametrization of the cuspidal cubic

I didn't understand why the method works fine to find the parametrization of the cuspidal curve: I didn't understand why finding these intersections points will give me the whole curve. thanks
user42912
  • 23,582
3
votes
1 answer

Is this set an algebraic set

Is the set $\{z\in\mathbb{C}:|z|=1\}$ and algebraic set? Intuitively, I think the answer is no because it is not possible to use a polynomial to split an arbitrary complex number into its real and imaginary components.
user44322
  • 993
3
votes
0 answers

Genus of a function field

There is a one-to-one correspondence between isomorphism classes of smooth absolutely irreducible curves $X/\mathbb{k}$ and isomorphism classes of fields $\mathbb{K}$ of transcendence degree $1$ over $\mathbb{k}$ such that $\mathbb{K}\cap…
Alexey Milovanov
  • 677
3
votes
1 answer

Why are smooth projective algebraic curves over the complex numbers compact and orientable?

Why smooth projective algebraic curves over complex numbers are compact and orientable? I want to use it to show algebraic smooth curves are topologically identical to toruses with different genuses. (projective algebraic curves are zeros of a…
Ahmadreza Khazaeipoul
  • 166
3
votes
2 answers

What is the equation for plotting points on a curve with fixed end points?

What is the equation for plotting points on an exponential curve with fixed end points? For example, if I want to plot 10 point along a curve that starts with 10,000 (x=1, y=10000) and ends with 30,000 (x=10, y=30000) the formula is…
Kuyenda
  • 141
3
votes
0 answers

Methods to prove that points do not lie on an algebraic plane curve

I have an infinite sequence of points in an affine plane and I want to show that these points do not lie on any algebraic plane curve. Are there any standard methods for doing this?
user55557
  • 31
1
2 3 4 5