0
votes
0answers
29 views

Questions on linear subspace of a projective space

I am a bit confused by the definition of the linear subspace of a projective space. It says in a book "Algebraic Geometry: A first course" by Joe Harris on page 5 that An inclusion of subspace ...
1
vote
0answers
58 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
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 ...
2
votes
1answer
86 views

$V(X^m + Y^m - Z^m)$ (projective Fermat curve) isomorphic to projective line iff $m=1, 2$

I've convinced myself that the projective Fermat curve $V(X^m + Y^m - Z^m) \subset \mathbb{P}^2$ is isomorphic to a projective line if and only if $m =1$ or $m = 2$, but I'm not sure how to prove this ...
2
votes
2answers
91 views

Space of global sections for a smooth projective curve of genus $g$

Let $X$ be a smooth projective curve of genus $g$, $T_{X}$ its tangent bundle and $H^{0}(X,T_{X})$ the space of global sections for $X$. What is $\dim H^{0}(X,T_{X})$ and why?
4
votes
1answer
106 views

Embedding of curves in projective spaces… typo?

I'm reading from the book "Geometry of algebraic curves", by Griffiths, Harris, Arbarello and Cornalba. In the middle of page 5 they define the map $\phi_{\mathscr{D}}:C\to \mathbb{P}V^*$, from a ...
4
votes
1answer
78 views

plane cubic with a singularity must have non-constant morphism from $\mathbb{P}^1$?

If $C$ is a plane projective curve which is defined by an irreducible homogeneous cubic polynomial and has a singularity, why must there be a nonconstant morphism $\mathbb{P}^1\rightarrow C$? (I'm ...
4
votes
1answer
140 views

The canonical divisor of the projective line

Let $A$ be a ring. Assume it has some nice properties if necessary, e.g., $A$ is a Dedekind domain. Let $\mathbf{P}^1_A$ be the projective line over $A$. I want to show that $-2 [\infty]$ is a ...
2
votes
0answers
125 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
267 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 $\mathbb{RP}^3$? ...
1
vote
0answers
104 views

Deducing characteristics of a map induced by a divisor

Given a divisor $D$ on an algebraic curve $X$, there is a corresponding map $\phi_D$ from $X$ to the projective space (of dimension $\dim L(D)-1$). In particular, we know that if $D$ is a very ample ...