Questions tagged [flips-flops]
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15 questions
2 votes
1 answer
238 views
Sequence of MMP with scaling cannot be isomorphism
Let $(X,B)$ be a projective klt pair, H is an $\mathbb{R}$-divisor s.t. $K_X+B+H$ is nef. Suppose that we can run $(K_X+B)$-MMP with scaling $H$(that is, the flip exists) and denote $\alpha_i:X \...
- 21
1 vote
0 answers
140 views
Canonical covering stack of a flop
In section 6 of Kawamata's paper https://arxiv.org/abs/math/0205287, he defines the canonical covering stack. In the proof of theorem 6.5, he considered a flop $$X\xrightarrow{\phi}W\xleftarrow{\psi}Y$...
- 439
7 votes
1 answer
443 views
Is the elementary transformation of a conic bundle a flip or a flop
Let $\pi: V\to S$ be a standard conic bundle of a threefold $V$ to a surface $S$, i.e., $\pi$ is relative minimal. Assume that everything is nonsingular and is over $\mathbb{C}$. We may assume that $V$...
- 165
3 votes
1 answer
388 views
Normal bundle and small contraction in threefolds
Let $f:X \to \mathbb{A}^1$ be a smooth, projective morphism of relative dimension $2$. Suppose that the fiber $X_0:=f^{-1}(0)$ contains an irreducible rational curve, say $C$ such that the restriction ...
- 2,062
1 vote
0 answers
143 views
Calculate amount of FLOPs for an eigenvalue problem solver
I've got 2 complex, non symmetric, matrices $A_{1000x1000}$, $B_{1000x1000}$ and I am using Matlab to get it's eigenvalues (functions like eig or eigs). Both matrices are different - one is more dense ...
- 111
2 votes
1 answer
293 views
Flipping and flipped loci
Let $f:X\dashrightarrow Y$ be the flip of a small contraction $\phi:X\rightarrow Z$, and let $\psi:Y\rightarrow Z$ be the small contraction such that $\psi\circ f = \phi$. Let $Exc(\phi), Exc(\psi)$ ...
user117617
3 votes
0 answers
225 views
Termination of flops vs termination of D-flops
It is known that any sequence of $D$-flops eventually terminates (in dimension at most four and with canonical singularities) where $D$ is an effective $\mathbb{Q}$-Cartier divisor which is negative ...
- 1,593
2 votes
0 answers
169 views
Stable base loci and flips
Let $D_1,D_2$ be two effective divisors on o normal and $\mathbb{Q}$-factorial projective variety $X$ of Picard rank two. Assume that $D_1$ is semi-ample and that it induces a small-comtraction $f_{...
- 9,090
2 votes
1 answer
224 views
Grassmannian inside a hyperkahler manifold
I am currently looking at stratified Mukai flop. Roughly speaking, this is a construction that, starting with a grassmannian $G$ inside a hyperkahler $X$ produces a birational manifold $X^*$ (with a ...
- 401
4 votes
1 answer
424 views
$Ex(f)$ has codimension at least 2
The following is a part of proof of lemma 6.2 in the book. $f:X \to Y$ a projective birational morphism of normal varieties $D$: Weil divisor on $Y$, $E$: exceptional divisor of $f$ $\mathcal{O}_X(...
- 41
9 votes
1 answer
540 views
Is the number of minimal models finite
Let $X$ be a variety of general type. Assume that $\dim X = 3$. In https://eudml.org/doc/164223 it is proven that $X$ has only finitely many minimal models (i.e., only $\mathbb Q$-factorial terminal ...
- 133
5 votes
1 answer
1k views
Recognizing a Mukai flop
Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose ...
- 527
12 votes
1 answer
2k views
Why is the standard flop a flop?
I have seen at least two ways to define flops (and similarly flips). We start with $Y \to X$, a surjective birational morphism, contracting a locus of codimension at least 2, such that $K_Y$ is ...
- 1,280
14 votes
0 answers
918 views
Is a flop on Calabi-Yau threefolds always Atiyah flop?
Is it true that any flop on a Calabi-Yau threefold is given by the Atiyah flop? That is, there always exists a rigid rational curve $\mathbb{P}^1$ with normal bundle $\mathcal{O}_{\mathbb{P}^1}(-1)^{\...
- 447
4 votes
1 answer
620 views
What is the Exceptional Locus of a flopping contraction between threefolds?
Hi, I'm trying to understand the group of cycles (modulo numerical equivalence) contracted by a flopping contraction $f$. More precisely, I'm in the setup of Definition 2.12 of this paper by ...
- 2,033