Algebraic Geometry - Raul Pandharipande - 2016

Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like finding out there's a video of yourself drunkenly break-dancing.

Romainb

He asked the following questions:

- Question 6 of the list (description of the Plücker embedding of \(Gr(2,4)\) in \(\mathbb{P}^5\) ).

- Is the quadric \( z_{12}z_{34}-z_{13}z_{24}+z_{14}z_{23}=0\) corresponding to \(Gr(2,4)\) in \(\mathbb{P}^5\) smooth?

- Is the Fermat curve \(z_{12}^2+z_{13}^2+z_{14}^2+z_{23}^2+z_{24}^2+z_{34}^2=0\) also isomorphic to \(Gr(2,4)\)? Is it smooth?

- What is the dimension of \(Gr(2,4)\)? What is the dimension of \(Gr(r,n)\)?

Then he gave me two curves \(x^4+y^2=0\) and \(x^4+17y^2=0\) in \( \mathbb{C}^2 \).

- What is the intersection multiplicity at \( (0,0) \)?

- Do they intersect in another point in \( \mathbb{C}^2 \)? In \( \mathbb{P}^2 \)?

- What is Bézout's Theorem?

Then Question 7 of the list i.e. "Is there a nonconstant regular map \( \mathbb{P}^2 \setminus \{p\} \longrightarrow \mathbb{C}^1 \)?"

22.08.2016 10.00-10.30 Nicola

Question 4

Intersection number of two cuspidal curves

Bézout --> where are the missing points, in the above? In projective space (find them)

Definition of Segre \( \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3 \)

Does a line in \(\mathbb{P}^3\) always intersect the image of Segre? (Yes, e.g. by dimension argument)

Dimension of space of lines tangent to image of Segre? (couldn't find it, should be 3)

Question 5