Algebraic Geometry - Raul Pandharipande - 2016

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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 \( \mathbb{P}^2 \setminus \{p\} \longrightarrow \mathbb{C}^1 \)?"