1
vote
0answers
44 views

On existence of a tangent line passing through a given point

Question Suppose $k$ is an algebraically closed field of characteristic $0$, and $C\subseteq\mathbb P^2(k)$ is an irreducible projective plane curve of degree $n>1$, and $P$ is a point on $\mathbb ...
2
votes
1answer
31 views

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 ...
1
vote
1answer
52 views

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 ...
2
votes
1answer
46 views

Calculating the projective closure with more than one generator

I am given a variety $X = Z(f_1,f_2)$ in affine 3-space (in $x,y,z$), and I would like to compute its projective closure $Y = Z(g_1,\dots,g_n)$ in projective 3-space (in $x,y,z,w$). I have seen this ...
1
vote
1answer
88 views

Prove that a curve in P^n of degree n not contained in a hyperplane is rational

The set up is as stated above. We have a projective curve $X$ of degree n embedded in $\mathbb{P^n}$, which is not contained in any hyperplane. We claim that it is therefore rational. The way I have ...
0
votes
0answers
74 views

Prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$

I'm meant to prove that irreducible curve of bidegree $(1, n)$ is rational for all $n \in \mathbb{N}$. I have a proof for this statement that uses the genus formula to show that such a curve must ...
2
votes
0answers
18 views

Getting used to projective coordinates, need help describing (2) objects geometrically [duplicate]

I'm trying to get an intuition for what things look like in projective coordinates. There are two curves that I have to work a problem with, but I'm not sure how to visualize them. They are $V(u^2 X ...
10
votes
2answers
270 views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
3
votes
2answers
189 views

Intersection of smooth projective plane curves

I need to calculate the number of intersections of the smooth projective plane curves defined by the zero locus of the homogeneous polynomials $$ F(x,y,z)=xy^3+yz^3+zx^3\text{ (its zero locus is ...
7
votes
1answer
342 views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus ...
5
votes
1answer
108 views

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set ...
2
votes
0answers
127 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
1
vote
0answers
271 views

What is the equation for a cone in $\mathbb{RP}^3$?

The zero locus of $x^2+y^2-z^2$ is a cone in $\mathbb{R}^3$. What is the projective version of this cone? That is, what is the homogeneous polynomial whose zero locus is a cone in $\mathbb{RP}^3$? ...
0
votes
2answers
171 views

some notions on algebraic curve

1) I want to learn about algebraic curves and i'm confused, please correct me if i'm wrong : when we say an Affine algebraic curve over the field $F$ : here affine to distinguish it from projective ...