Local Models of Standard Flips

• Local models of standard flips are explicit birational frameworks that isolate exceptional loci to enable concrete computations in flip geometry.
• They provide projective, normal-bundle, and VGIT descriptions that facilitate analyses in cohomology, quantum D-modules, and derived categories.
• Their realizations in Hilbert schemes and Fano varieties underscore a unified approach bridging categorical decompositions with quantum cohomology methods.

Local models of standard flips are explicit birational geometries that isolate the behavior of a flip along its exceptional locus and replace the ambient variety by a projective-bundle, normal-bundle, or GIT model in which the two contractions and the common exceptional divisor can be written down concretely. In the literature on simple $(r,r')$-flips, standard flips, generalized Grassmannian flips, and VGIT wall-crossings, the local model is the mechanism that makes the flip computable at the level of cohomology, quantum $D$-modules, Gamma classes, and derived categories (Lee et al., 2019, Shen et al., 12 Feb 2025, Leung et al., 2023, Gu et al., 21 Aug 2025, Shah, 7 Feb 2026).

1. Definition and basic birational structure

A flip is a birational map

$f:Y_2 \dashrightarrow Y_1$

if there exists a normal variety $W$ and small contractions $\phi_i:Y_i\to W$ such that $f=\phi_1^{-1}\circ \phi_2$, with $-K_{Y_2}$ $\phi_2$-ample and $K_{Y_1}$ $\phi_1$-ample. A flip is called simple if the two birational morphisms resolving $D$0 are blow-ups along smooth centers: $D$1 In this situation the exceptional divisor $D$2 in the resolution carries two projective bundle structures, one over each center (Leung et al., 2023).

The standard flip is the relative form used in later work on Hilbert schemes and quantum geometry. One starts with a smooth projective variety $D$3 containing a closed subvariety

$D$4

such that $D$5 is a projective bundle over a smooth base $D$6,

$D$7

and the normal bundle has the form

$D$8

for another vector bundle $D$9 on $f:Y_2 \dashrightarrow Y_1$0. Blowing up $f:Y_2 \dashrightarrow Y_1$1 in $f:Y_2 \dashrightarrow Y_1$2,

$f:Y_2 \dashrightarrow Y_1$3

produces an exceptional divisor

$f:Y_2 \dashrightarrow Y_1$4

where $f:Y_2 \dashrightarrow Y_1$5, and

$f:Y_2 \dashrightarrow Y_1$6

A second contraction

$f:Y_2 \dashrightarrow Y_1$7

contracts $f:Y_2 \dashrightarrow Y_1$8 in the other ruling to $f:Y_2 \dashrightarrow Y_1$9. The birational map $W$0 together with the common blow-up $W$1 is called a standard flip; if $W$2, it is a standard flop (Shah, 7 Feb 2026).

This formulation already exhibits the essential local model: the exceptional divisor is simultaneously a projective bundle over two different centers, and the normal bundle $W$3 governs the possibility of blowing down in the second direction. The case $W$4 or $W$5 is a blowup, and $W$6 is an ordinary flop (Gu et al., 21 Aug 2025).

2. Projective and normal-bundle local models

The projective local model of a simple $W$7-flip is completely explicit. The exceptional loci are

$W$8

with normal bundles

$W$9

The corresponding projective local models are

$\phi_i:Y_i\to W$0

They satisfy

$\phi_i:Y_i\to W$1

and $\phi_i:Y_i\to W$2 is Fano. The cohomology ring of $\phi_i:Y_i\to W$3 is written explicitly as

$\phi_i:Y_i\to W$4

with $\phi_i:Y_i\to W$5 the hyperplane class of $\phi_i:Y_i\to W$6 and $\phi_i:Y_i\to W$7 the tautological divisor class on $\phi_i:Y_i\to W$8 (Lee et al., 2019).

A relative version replaces projective spaces by bundles over a smooth base $\phi_i:Y_i\to W$9. Here the exceptional locus is

$f=\phi_1^{-1}\circ \phi_2$0

and the normal bundle in $f=\phi_1^{-1}\circ \phi_2$1 has the form

$f=\phi_1^{-1}\circ \phi_2$2

for a vector bundle $f=\phi_1^{-1}\circ \phi_2$3 of rank $f=\phi_1^{-1}\circ \phi_2$4, while $f=\phi_1^{-1}\circ \phi_2$5 has rank $f=\phi_1^{-1}\circ \phi_2$6, with $f=\phi_1^{-1}\circ \phi_2$7. The exceptional divisor in the blow-up is

$f=\phi_1^{-1}\circ \phi_2$8

with normal bundle $f=\phi_1^{-1}\circ \phi_2$9. Contracting this divisor in the smooth algebraic-space category gives the flipped variety $-K_{Y_2}$0 (Shen et al., 12 Feb 2025).

The local model is then defined intrinsically from $-K_{Y_2}$1 by the $-K_{Y_2}$2-action on $-K_{Y_2}$3 that scales $-K_{Y_2}$4 with weight $-K_{Y_2}$5 and $-K_{Y_2}$6 with weight $-K_{Y_2}$7. The GIT quotients for characters of weights $-K_{Y_2}$8 and $-K_{Y_2}$9 give two spaces

$\phi_2$0

identified as

$\phi_2$1

with a rational map $\phi_2$2. When $\phi_2$3 is a genuine flip, one may identify $\phi_2$4 with $\phi_2$5 and $\phi_2$6 with $\phi_2$7 (Shen et al., 12 Feb 2025).

A recurring point in the literature is that the projective local model and the normal-bundle local model are not competing descriptions. The former is the explicit absolute model used for analytic continuation and $\phi_2$8-manifold calculations, while the latter packages the same geometry relative to a base and is the form needed for quantum and categorical decompositions. This suggests that “local model” names a precise replacement geometry, not merely a heuristic neighborhood description.

3. VGIT and homogeneous realizations

Simple VGIT wall-crossings provide a systematic source of local models for standard flips. For a smooth quasi-projective variety $\phi_2$9 with a linear action of a reductive group $K_{Y_1}$0, a simple wall-crossing

$K_{Y_1}$1

arises from adjacent chambers in the GIT-effective cone. The wall geometry has the form of a standard flip: the exceptional loci $K_{Y_1}$2 are projective bundles over a common wall $K_{Y_1}$3, with normal bundles

$K_{Y_1}$4

and the blowups of $K_{Y_1}$5 and $K_{Y_1}$6 along $K_{Y_1}$7 and $K_{Y_1}$8 are canonically identified with the fiber product $K_{Y_1}$9 (Gu et al., 21 Aug 2025).

In birational terms, the local model of such a flip is obtained by replacing $\phi_1$0 with the total spaces of the normal bundles of the exceptional loci. The same paper emphasizes that these local models can themselves be realized as a simple VGIT wall-crossing for an appropriate $\phi_1$1-action, so the standard flip local model is not external to VGIT but a concrete instance of it (Gu et al., 21 Aug 2025).

A second generalization replaces projective spaces and ordinary Grassmannians by homogeneous spaces. A simple flip is of homogeneous type if the exceptional divisor $\phi_1$2 is a homogeneous variety; if $\phi_1$3, it is of rank $\phi_1$4; if $\phi_1$5, it is called a generalized Grassmannian flip. In the construction from a marked Dynkin diagram $\phi_1$6, one has a homogeneous space $\phi_1$7 and projections

$\phi_1$8

where $\phi_1$9 and $D$00 are obtained by removing one marked node from $D$01. Under Condition (A), the general fibers of both $D$02 and $D$03 are projective spaces. One then defines

$D$04

proves that $D$05 is a vector bundle on $D$06, and obtains

$D$07

The ambient spaces are

$D$08

and the blow-ups

$D$09

produce the flip $D$10 (Leung et al., 2023).

This homogeneous framework contains the classical standard flip: for type $D$11, the case $D$12 is the standard flip (Leung et al., 2023). A plausible implication is that the phrase “local model of a standard flip” now encompasses a hierarchy: projective local models, relative normal-bundle models, VGIT models, and homogeneous models built from $D$13.

4. Quantum cohomology and quantum $D$14-modules

The local model first became a computational device for quantum cohomology in the study of analytic continuation along the extremal ray. For a simple $D$15-flip, the extremal curve class on $D$16 is denoted $D$17, with Novikov variable $D$18, and the two sides are related by

$D$19

The graph-closure correspondence

$D$20

identifies the Chow motive of $D$21 with a submotive of that of $D$22 and preserves the Poincaré pairing, but it does not preserve the cup product in general. The local-model analysis constructs a deformation $D$23 of $D$24 and produces a non-linear embedding

$D$25

in the category of $D$26-manifolds into the regular integrable loci of $D$27 near $D$28 (Lee et al., 2019).

In the relative standard-flip setting, the quantum theory is restricted to the exceptional curve direction. If $D$29 is a line in a fiber of $D$30, one keeps only degrees $D$31: $D$32 This yields the exceptional quantum product $D$33. A central reduction theorem gives a ring homomorphism

$D$34

where $D$35 is the local model. The exceptional quantum cohomology ring of $D$36 is presented as

$D$37

with $D$38 the Chern roots of $D$39, $D$40 those of $D$41, and

$D$42

(Shen et al., 12 Feb 2025).

The same local model controls the $D$43-function and its asymptotics. After rewriting the relevant expressions with Gamma functions, the asymptotic analysis is reduced to Meijer $D$44-functions. The paper uses two regimes: nonzero eigenvalues, governed by $D$45, and zero eigenvalues, governed by a $D$46-type function. The outcome is a decomposition of the quantum cohomology of standard flips into asymptotic Gamma classes (Shen et al., 12 Feb 2025).

VGIT yields a complementary quantum description. For a simple $D$47-VGIT wall-crossing with $D$48, after a suitable formal change of variables there is an isomorphism of quantum $D$49-modules

$D$50

compatible with both the quantum connections and the pairings. The paper states this as an application to the local models of standard flips in birational geometry. When $D$51, the wall-crossing is crepant or ordinary flop type, and the theorem reduces to invariance after analytic continuation (Gu et al., 21 Aug 2025).

These results address a common misconception: the local model is not only a way to visualize the exceptional locus. In the cited works it supports explicit Picard–Fuchs operators, Dubrovin connections, $D$52-functions, quantum ring presentations, and direct-sum decompositions of quantum $D$53-modules.

5. Derived categories and Gamma compatibility

Derived-category applications are built into the definition of the standard flip. For $D$54, the standard flip induces a semiorthogonal decomposition

$D$55

where

$D$56

This is one of the central categorical consequences of the standard local model (Shah, 7 Feb 2026).

Generalized Grassmannian flips exhibit the same pattern in a homogeneous setting. Using Orlov’s blow-up formula, one obtains semiorthogonal decompositions of $D$57 from the two contractions. In the $D$58 example, the 9-folds

$D$59

are related by a flip

$D$60

their common blow-up is

$D$61

and the common exceptional divisor is

$D$62

By mutations in the sense of Kuznetsov and Thomas’ chess game method, the paper proves

$D$63

thereby verifying the DK flip conjecture in this case (Leung et al., 2023).

The Gamma-theoretic side is compatible with these semiorthogonal decompositions. For standard flips, the asymptotic decomposition of quantum cohomology is matched with the Orlov and Belmans–Fu–Raedschelders semiorthogonal decomposition component-by-component. On the cohomological side this is expressed by

$D$64

where the maps $D$65 and $D$66 are built from Todd classes (Shen et al., 12 Feb 2025).

A plausible implication is that the phrase “local model” has become categorical as well as geometric: once the local flip geometry is explicit, semiorthogonal decompositions, fully faithful functors, and Gamma-class asymptotics can be compared on the same model.

6. Explicit geometric realizations on Hilbert schemes of quadrics

The local model of the standard flip is especially concrete in the geometry of Hilbert schemes of quadrics. For a smooth cubic hypersurface $D$67, the classical birational map between the Hilbert square $D$68 and a projective bundle over the Fano scheme of lines is identified as a standard flip. In the case $D$69, the center on the $D$70-side is

$D$71

the opposite center is $D$72, and the exceptional divisor is a $D$73-bundle over $D$74. For a line $D$75, the normal bundle computation

$D$76

is the local input that yields the standard flip structure (Shah, 7 Feb 2026).

The same mechanism extends to del Pezzo varieties of degree at least $D$77. If $D$78 is an $D$79-dimensional del Pezzo variety with degree

$D$80

then $D$81 is very ample and there is a standard flip diagram

$D$82

with $D$83 smooth projective. Here $D$84 is the relative Hilbert scheme of $D$85 points on the universal line over $D$86 (Shah, 7 Feb 2026).

For complete intersections of two quadrics,

$D$87

the flipped side is a relative orthogonal Grassmannian $D$88, where $D$89 is the quadric fibration associated with the pencil $D$90. For linear sections of $D$91,

$D$92

the flipped side is a relative Grassmannian

$D$93

In each case the local construction is controlled by span geometry and by a section

$D$94

which encodes the residual quadric obtained from the defining equations (Shah, 7 Feb 2026).

These examples clarify two boundary cases. First, a standard flip is not simply any birational correspondence between moduli spaces; it requires the projective-bundle center and the normal bundle of the standard form. Second, the local model is not restricted to abstract birational theory: it appears concretely in Hilbert schemes, Fano varieties of linear spaces, orthogonal Grassmannians, and linear sections of homogeneous varieties. In this sense, the local model of a standard flip is the common geometric template behind explicit birational transformations, categorical decompositions, and quantum-structural calculations.