--- title: Local Models of Standard Flips url: https://www.emergentmind.com/topics/local-models-of-standard-flips type: topic --- # Local Models of Standard Flips 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 [1912.03012], [2502.08762], [2309.11136], [2508.15770], [2602.07366]. ## 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 \(f\) are blow-ups along smooth centers: \[ f = \pi_1 \circ \pi_2^{-1}, \qquad X \xrightarrow{\pi_2} Y_2,\quad X \xrightarrow{\pi_1} Y_1. \] In this situation the exceptional divisor \(E\) in the resolution carries two projective bundle structures, one over each center [2309.11136]. The standard flip is the relative form used in later work on Hilbert schemes and quantum geometry. One starts with a smooth projective variety \(X\) containing a closed subvariety \[ \iota: Z \hookrightarrow X \] such that \(Z\) is a projective bundle over a smooth base \(F\), \[ Z \simeq \mathbf{P}_F(E), \] and the normal bundle has the form \[ N_{Z/X} \simeq \mathcal O_\pi(-1)\otimes \pi^*E' \] for another vector bundle \(E'\) on \(F\). Blowing up \(X\) in \(Z\), \[ \phi:\widetilde X:=\mathrm{Bl}_Z X \to X, \] produces an exceptional divisor \[ E \simeq \mathbf{P}_Z(\pi^*E') \simeq \mathbf{P}_{Z'}(\pi^*E), \] where \(Z'=\mathbf{P}_F(E')\), and \[ N_{E/\widetilde X}\simeq \mathcal O_E(-1,-1). \] A second contraction \[ \phi':\widetilde X\to X' \] contracts \(E\) in the other ruling to \(Z'\subset X'\). The birational map \(X \dashleftrightarrow X'\) together with the common blow-up \(\widetilde X\) is called a standard flip; if \(\mathrm{rank}(E)=\mathrm{rank}(E')\), it is a standard flop [2602.07366]. This formulation already exhibits the essential local model: the exceptional divisor is simultaneously a projective bundle over two different centers, and the normal bundle \(\mathcal O(-1,-1)\) governs the possibility of blowing down in the second direction. The case \(r_+=1\) or \(r_-=1\) is a blowup, and \(r_+=r_-\) is an ordinary flop [2508.15770]. ## 2. Projective and normal-bundle local models The projective local model of a simple \((r,r')\)-flip is completely explicit. The exceptional loci are \[ Z \cong \mathbb P^r \subset X, \qquad Z' \cong \mathbb P^{r'} \subset X', \] with normal bundles \[ N_{Z/X} \cong \mathcal O_{\mathbb P^r}(-1)^{\oplus (r'+1)}, \qquad N_{Z'/X'} \cong \mathcal O_{\mathbb P^{r'}}(-1)^{\oplus (r+1)}. \] The corresponding projective local models are \[ X_{\mathrm{loc}} = \mathbb P_{\mathbb P^r}\!\bigl(\mathcal O(-1)^{\oplus (r'+1)} \oplus \mathcal O\bigr), \qquad X'_{\mathrm{loc}} = \mathbb P_{\mathbb P^{r'}}\!\bigl(\mathcal O(-1)^{\oplus (r+1)} \oplus \mathcal O\bigr). \] They satisfy \[ \dim X_{\mathrm{loc}}=\dim X'_{\mathrm{loc}}=r+r'+1, \] and \(X_{\mathrm{loc}}\) is Fano. The cohomology ring of \(X\) is written explicitly as \[ H(X)=\mathbb Z[h,\xi]/(h^{r+1},\ \xi(\xi-h)^{r'+1}), \] with \(h\) the hyperplane class of \(\mathbb P^r\) and \(\xi\) the tautological divisor class on \(X\to \mathbb P^r\) [1912.03012]. A relative version replaces projective spaces by bundles over a smooth base \(Z\). Here the exceptional locus is \[ F=\mathbf{P}(V)\to Z, \] and the normal bundle in \(X\) has the form \[ N_{F|X}=T^*V' \otimes \mathcal O_{\mathbf{P}(V)}(-1) \] for a vector bundle \(V'\to Z\) of rank \(s\), while \(V\to Z\) has rank \(r\), with \(r>s\). The exceptional divisor in the blow-up is \[ E \cong \mathbf{P}(V)\times_Z \mathbf{P}(V') \] with normal bundle \(\mathcal O(-1,-1)\). Contracting this divisor in the smooth algebraic-space category gives the flipped variety \(X'\) [2502.08762]. The local model is then defined intrinsically from \((Z,V,V')\) by the \(\mathbb C^*\)-action on \(\mathrm{tot}(V\oplus V')\) that scales \(V\) with weight \(+1\) and \(V'\) with weight \(-1\). The GIT quotients for characters of weights \(+1\) and \(-1\) give two spaces \[ T,\qquad T', \] identified as \[ T=\mathrm{tot}\bigl(V'(-1)\bigr)=\mathrm{tot}\bigl(T^*V'\otimes \mathcal O_{\mathbf P(V)}(-1)\bigr), \qquad T'=\mathrm{tot}\bigl(V(-1)\bigr), \] with a rational map \(T\dashrightarrow T'\). When \(X\to X'\) is a genuine flip, one may identify \(T\) with \(\mathrm{tot}(N_{F|X})\) and \(T'\) with \(\mathrm{tot}(N_{F'|X'})\) [2502.08762]. 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 \(F\)-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 \(W\) with a linear action of a reductive group \(G\), a simple wall-crossing \[ X_- \dashrightarrow X_+ \] arises from adjacent chambers in the GIT-effective cone. The wall geometry has the form of a standard flip: the exceptional loci \(P_\pm\subset X_\pm\) are projective bundles over a common wall \(S\), with normal bundles \[ N_{P_\pm/X_\pm}\cong \psi_\pm^*(V_\mp)(-1), \] and the blowups of \(X_-\) and \(X_+\) along \(P_-\) and \(P_+\) are canonically identified with the fiber product \(X_+\times_{X_0}X_-\) [2508.15770]. In birational terms, the local model of such a flip is obtained by replacing \(X_\pm\) 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 \(\mathbb C^*\)-action, so the standard flip local model is not external to VGIT but a concrete instance of it [2508.15770]. A second generalization replaces projective spaces and ordinary Grassmannians by homogeneous spaces. A simple flip is of homogeneous type if the exceptional divisor \(E\) is a homogeneous variety; if \(p(E)=r+1\), it is of rank \(r\); if \(r=1\), it is called a generalized Grassmannian flip. In the construction from a marked Dynkin diagram \((T,I)\), one has a homogeneous space \(G/P_I\) and projections \[ p_a: G/P_I \to G/P_{I_a}, \qquad p_b: G/P_I \to G/P_{I_b}, \] where \(I_a\) and \(I_b\) are obtained by removing one marked node from \(I\). Under Condition (A), the general fibers of both \(p_a\) and \(p_b\) are projective spaces. One then defines \[ E_i := p_{i*}\big(\mathcal{O}(p_a h_a + p_b h_b)\big), \] proves that \(E_i\) is a vector bundle on \(G/P_{I_i}\), and obtains \[ G/P_I \cong \mathbb{P}(E_i). \] The ambient spaces are \[ X_i := \mathbb{P}_{G/P_{I_i}}(E_i^\vee \oplus \mathcal O), \qquad i=a,b, \] and the blow-ups \[ \mathrm{Bl}_{Z_a}X_a \cong \mathrm{Bl}_{Z_b}X_b =: X \] produce the flip \(f:X_a \dashrightarrow X_b\) [2309.11136]. This homogeneous framework contains the classical standard flip: for type \(A_m \times A_n\), the case \((\alpha_1,\beta_1)\) is the standard flip [2309.11136]. 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 \(G/P\). ## 4. Quantum cohomology and quantum \(D\)-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 \((r,r')\)-flip, the extremal curve class on \(X\) is denoted \(\ell\), with Novikov variable \(q^\ell\), and the two sides are related by \[ q^\ell = \frac{1}{q^{\ell'}}. \] The graph-closure correspondence \[ [\bar\Gamma_f]^*: H(X') \longrightarrow H(X) \] identifies the Chow motive of \(X'\) with a submotive of that of \(X\) and preserves the Poincaré pairing, but it does not preserve the cup product in general. The local-model analysis constructs a deformation \(\widehat\Psi\) of \(\Psi=[\Gamma_f]^*\) and produces a non-linear embedding \[ QH(X') \hookrightarrow QH(X) \] in the category of \(F\)-manifolds into the regular integrable loci of \(QH(X)\) near \(q^\ell = \infty\) [1912.03012]. In the relative standard-flip setting, the quantum theory is restricted to the exceptional curve direction. If \(L\subset F\) is a line in a fiber of \(F=\mathbf P(V)\to Z\), one keeps only degrees \(k[L]\): \[ Q_d= \begin{cases} 1 & d=k[L],\ k\ge 0,\\ 0 & \text{otherwise}. \end{cases} \] This yields the exceptional quantum product \(*_{q,t}\). A central reduction theorem gives a ring homomorphism \[ i^*:\mathrm{QH}_{\mathrm{exc}}(X)\to \mathrm{QH}_{\mathrm{exc}}(T), \] where \(T\) is the local model. The exceptional quantum cohomology ring of \(T=\mathrm{tot}(V'(-1))\) is presented as \[ \mathrm{QH}_{\mathrm{exc}}(T)=H^*(Z)[H][[q]]\Big/\left(\prod_{i=1}^r (p_i+H)=q^{r-s}\prod_{j=1}^s (o_j-H)\right), \] with \(p_i\) the Chern roots of \(V\), \(o_j\) those of \(V'\), and \[ H=c_1\bigl(\mathcal O_{\mathbf P(V)}(1)\bigr) \] [2502.08762]. The same local model controls the \(J\)-function and its asymptotics. After rewriting the relevant expressions with Gamma functions, the asymptotic analysis is reduced to Meijer \(G\)-functions. The paper uses two regimes: nonzero eigenvalues, governed by \(G_{r,0}^{0,r}\), and zero eigenvalues, governed by a \(G_{r,1}^{1,r}\)-type function. The outcome is a decomposition of the quantum cohomology of standard flips into asymptotic Gamma classes [2502.08762]. VGIT yields a complementary quantum description. For a simple \(G\)-VGIT wall-crossing with \(r_+