7
votes
Accepted
Is the elementary transformation of a conic bundle a flip or a flop
It is neither flip nor flop, because the exceptional locus both on $V$ and on $V'$ is a divisor.
- 37.3k
4
votes
Accepted
Flipping and flipped loci
I don't think so. In general we always have
$$
\dim \text{Exc}\phi+\dim \text{Exc}\psi\geq \dim X-1.
$$
This is proved in Lemma 5-1-7 [Kawamata, Matsuda, Matsuki, Introduction to Minimal Model Program]...
- 1,084
3
votes
Accepted
Normal bundle and small contraction in threefolds
The exact sequence
$$
0 \to N_{C/X_0} \to N_{C/X} \to N_{X_0/X}\vert_C \to 0
$$
in this case reads as
$$
0 \to \mathcal{O}_C(-2) \to N_{C/X} \to \mathcal{O}_C \to 0
$$
which means that either $(a,b) = ...
- 37.3k
3
votes
Accepted
Grassmannian inside a hyperkahler manifold
This is just the general case of the example from my comment above. Let $S\subset \mathbb{P}^g_{\mathbb{C}}$ be a K3 surface of degree $2g-2$ that contains no curve that spans a $\mathbb{P}^r$ with $...
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