Questions tagged [flips-flops]

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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$...
7
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1answer
313 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 $V$...
3
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1answer
168 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 ...
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65 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 ...
3
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1answer
161 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 $Exc(\phi), Exc(\psi)$ ...
3
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163 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 ...
3
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138 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_{...
2
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1answer
157 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 ...
4
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1answer
227 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(...
9
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1answer
400 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 ...
4
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1answer
828 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 ...
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1answer
1k 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 ...
14
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0answers
720 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)^{\...
4
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1answer
435 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 ...