2
votes
1answer
60 views

Smooth surface that is a complete intersection

I have this definition of a projective complex algebraic surface that is a complete intersection. A surface $S\subset\mathbb{P}^{r+2}$ is said to be a complete intersection if it is a trasversal ...
1
vote
0answers
94 views

Transversal and complete intersection of hypersurfaces in $\mathbb{P}^{n}$

(a) Let $k<n$ and $F_{1},\dots,F_{k}$ be homogeneous polynomials of degrees $d_{1},\dots,d_{k}$ of $n+1$ variables in generic case. Prove that the corresponding hypersurfaces in ...
3
votes
1answer
79 views

Projective lines which can be viewed in some sense as surfaces

The complex projective line can be viewed as the $2$-sphere. I'd appreciate some examples of other projective lines (over any field or even ring) that can be viewed in some sense as a surface. I am ...
2
votes
2answers
352 views

projection of a quadric surface

Consider the quadric surface $X = \{ xy = zw \} \subset \mathbb{P}^3$ and pick a point $x \in X$. I think it is true that if we think of $\mathbb{P}^2$ as the space of lines through $x$ in ...
4
votes
0answers
202 views

Hirzebruch surfaces

How can I express the 2nd Hirzebruch surface, $F_{2}$ in terms of $SO(3)$. Is it true that $F_{2}$ is the total space of a bundle with fibre SO(3) over $\mathbb{R}_{+}$?