4

I expect that this is a very easy question, but somehow I can't get it. What is the dimension of the moduli space of complete intersections of degree 2 and 4 in $\mathbb{P}^5$? The answer should be 89.

I apologize again that if the question is too easy.

user148177
  • 1,315

1 Answers1

1

Here's how I arrived at my answer.

First, choose a quadric in $\mathbf P^5$. The parameter space for such choices has dimension ${5+2 \choose 2} -1$ = 20.

Next, choose a quartic in $\mathbf P^5$, modulo the already chosen quadric. (Two quartics forms which differ by a multiple of the chosen quadric will give the same complete intersection.) The parameter space for such choices has dimension $ \left( {5+4 \choose 4} - {5+2\choose 2} \right) -1$ = 126-21-1=104.

OK, so the parameter space of complete intersections has dimension 104+20=124. But now we want the moduli space, so we divide out by the action of $PGL(6)$. That has dimension $6^2-1=35$. So in the end we get a space of dimension $124-35=89$.