Flip (algebraic geometry)¶
A small birational replacement over one base that turns a negative relative canonical or log-canonical direction into a positive one without contracting a divisor.
Core Idea¶
A flip replaces a small birational contraction that is negative for a canonical or log-canonical divisor with a second small model over the same base where the transformed divisor is positive. Neither side contracts a divisor, so the models agree in codimension one while their exceptional loci change.[ref-987cd30e5251][ref-43c370862060] A negative small contraction without its positive replacement is only a flipping contraction. A K-trivial replacement is a flop, not this negative-to-positive move.[^ref-4d278e639797]
Scope of Application¶
Flips occur in birational geometry when a negative small contraction has a positive replacement.
- Toric threefolds: fan subdivisions give explicit K-negative and K-positive small maps over one base.[^ref-987cd30e5251]
- Log moduli models: an elliptic-bridge contraction has a small Mori flip over a Chow quotient.[^ref-43c370862060]
- Ordinary K-flips: the controlling boundary Δ is zero, so the direction is set by K alone.[^ref-987cd30e5251]
- Log flips: a fixed K+Δ tests the relative sign on both birational models.[^ref-43c370862060]
- MMP steps: one flip may advance a program; proving whole-program termination is separate.[^ref-dcc682df39bc]
Clarity¶
The three tests are smallness, one base, and relative sign reversal. Exchanging curves without deleting a divisor is insufficient if the canonical divisor is K-trivial; the Atiyah example is a flop.[^ref-4d278e639797] A divisorial contraction fails because it removes a divisor. In the stable-curve moduli example, represented elliptic-bridge and tacnodal curves describe the affected loci, while the flip transforms the ambient moduli spaces.[^ref-43c370862060] The positive sign is a mathematical intersection condition, not a guarantee of smoothness.
Manages Complexity¶
A construction may be written with toric fans, singularity conditions or GIT quotients. The common diagram reduces it to an input model, a negative small map, a shared target, a positive small map, and codimension-one agreement. Fujino and Sato verify the signs through a toric parameter inequality; their maps are projective over the common base, without asserting absolute projectivity.[^ref-987cd30e5251] Hassett and Hyeon construct a flipped Hilbert-quotient model over a Chow-quotient base.[^ref-43c370862060] These proofs use different tools but establish the same five roles.
Abstract Reasoning¶
For any claimed flip, hold one canonical or log-canonical divisor fixed while comparing both sides. Check that the input contraction is small and negative, the replacement map is small and positive over the same base, and divisorial data persist. This distinguishes a completed flip from a flop, divisorial contraction or proposed replacement with no positive model. In the moduli case, the input model's “+ε” label must not be mistaken for the fixed K+(7/10−ε)Δ used to test both signs.[^ref-43c370862060]
Knowledge Transfer¶
The identity test transfers between an ordinary toric K-flip and a log flip of stable-curve moduli: the objects, constructions and exceptional loci differ, while the negative-to-positive small diagram persists.[ref-987cd30e5251][ref-43c370862060] The broader Prime Transformation supplies input, rule, output, preserved divisorial data and altered exceptional structure; the canonical-divisor test keeps this entry domain-specific. A flip is one possible move of the Minimal Model Program, not the whole program. Existence of klt flips under BCHM's stated hypotheses does not establish unrestricted termination of sequences.[^ref-dcc682df39bc]
Relationships to Other Abstractions¶
Current abstraction Flip (algebraic geometry) Domain-specific
Parents (1) — more general patterns this builds on
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Flip (algebraic geometry) is a kind of Transformation Prime
Every flip is a rule-governed birational transformation that preserves codimension-one data while replacing a small exceptional locus and reversing the relative canonical or log-canonical sign.
Hierarchy path (1) — routes to 1 parentless root
- Flip (algebraic geometry) → Transformation → Function (Mapping)
Neighborhood in Abstraction Space¶
Flip (algebraic geometry) sits in a sparse region of the domain-specific corpus (92nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2721 abstractions)
Nearest neighbors
- Minimal Model Program — 0.83
- Logarithmic pair — 0.79
- Contraction Morphism — 0.78
- Logarithmic Form — 0.78
- Jacobian variety — 0.78
Computed from structural-signature embeddings · 2026-10-17
References¶
[^ref-987cd30e5251]: Osamu Fujino and Hiroshi Sato, Notes on toric varieties from Mori theoretic viewpoint, II, Nagoya Mathematical Journal 239 (2020), 42–75, §3.3 Example 3.3.2 and Remark 3.3.3. Original full-text construction; projective over the common base. [^ref-43c370862060]: Brendan Hassett and Donghoon Hyeon, Log minimal model program for the moduli space of stable curves: The first flip, arXiv:0806.3444v1 (2008), Abstract, §1, Proposition 2.9, Theorem 2.12, Corollary 2.15 and §9 proof. Original full-text preprint; g≥4 and fixed K+(7/10−ε)Δ. [^ref-dcc682df39bc]: Caucher Birkar, Paolo Cascini, Christopher D. Hacon and James McKernan, Existence of Minimal Models for Varieties of Log General Type, arXiv:math/0610203, §1.4 Corollary 1.4.1 and following termination qualification. Original full-text theorem in its klt setting. [^ref-4d278e639797]: Masayuki Kawakita, Complex Algebraic Threefolds, Chapter 1, Example 1.3.18, publisher full-chapter preview pp. 38–39. Specialist negative-boundary account of the K-trivial Atiyah flop.