Global CY threefold flop model from perverse-point Quot schemes

Rationality directly from crepant duality. Normality and proper birationality give $f\ast OV=R$: a rational function regular on V has no pole along any height-one prime of R, and normality identifies R with their intersection. By [F7], the geometric fiber bound gives $R^{i}f\ast OV=0$ for $i\ge 2$; the potentially nonzero $L=R^{1}f\ast OV$ is finite length since f is an isomorphism off m. Put $C=Rf\ast OV$, so there is a truncation triangle $R\to C\to L[-1]$.

Work at one of the finitely many closed exceptional-image points, with $R=O_{X,x}$. Choose a regular local C-algebra A of dimension d and a surjection $A\to R$ from a local affine-space presentation. The closed immersion Spec $R\to Spec$ A and the projectivity of f make $V_R$ projective over A, so the regular-base supplier applies to $V_R$ over A. Its dualizing complex is $D_V={\omega }_{V_R}[3]$, by the smooth local canonical calculation in that supplier; the quotient supplier gives $D_R=RHo{m}_A(R,A[d])$. The Gorenstein hypothesis identifies $D_R={\omega }_R[3]$. Put $C=R\mathrm{\Gamma }(V_R,O_V)$, a bounded finite R-complex by F2 and F7. Derived coinduction gives $RHo{m}_R(C,D_R)=RHo{m}_A(C,A[d])$: take an injective A-resolution I of A[d], observe ${\mathrm{Hom}}_A(R,I)$ is termwise R-injective, and use ${\mathrm{Hom}}_R(C,{\mathrm{Hom}}_A(R,I))={\mathrm{Hom}}_A(C,I)$. Regular-base projective duality now identifies this with $R\mathrm{\Gamma }(V_R,D_V)$. Crepancy and projection formula therefore give $RHo{m}_R(C,{\omega }_R[3])=C\otimes {\omega }_R[3]$; trivializing the invertible ${\omega }_R$ yields $RHo{m}_R(C,R)=C$. This argument precedes and does not use rationality. For finite-length L, the same coinduction computes $RHo{m}_R(L,{\omega }_R[3])=RHo{m}_A(L,A[d])$, concentrated in degree zero: the Koszul resolution of the residue field gives the assertion for that field, and a finite composition series gives it for L. Denote this exact finite-length dual by L∨; biduality gives $L\vee =0$ only when $L=0$. Thus $RHo{m}_R(L,R)=L\vee [-3]$, after the same canonical-line trivialization. Dualizing the truncation triangle gives $L\vee [-2]\to C\to R\to L\vee [-1]$.

In degree one this triangle forces $H^1(C)=0$: the degree-one cohomology of its left and right terms is zero. Its degree-two segment also forces $L\vee =0$. Consequently $C=R$. No rational-singularity theorem or nef-big vanishing has been assumed. The use of coherent duality here is the actual finite-map/regular-embedding duality, obtainable from a regular local presentation of R and a projective embedding of V; it is not an assumption that V has a tilting object.

The regular-presentation construction in step 2.1 applies to every bounded coherent F on $V_R$, not only $O_V$: regular-base projective duality and the identical coinduction calculation give $RHo{m}_R(R\mathrm{\Gamma }(V_R,F),D_R)=R\mathrm{\Gamma }(V_R,RHom(F,D_V))$. By crepancy this is precisely relative coherent duality with $f^{!}{O}_X=O_V$, after cancelling the canonical line and shift. The comparisons are the canonical trace/evaluation comparisons of the supplied embedding, so agree under restriction and glue. For families used below, the finite locally free relative kernel models and bounded Cech totalizations over C give their base-change trace maps; no application of a regular-base assertion directly to the singular R is made.

The projective Quot construction. Let V,S be projective complex schemes and E a coherent sheaf on V×S. Fix a very ample line bundle H on V and a fiber Hilbert polynomial P. The functor of finitely presented S-flat quotients of E with polynomial P is represented by a projective S-scheme, as follows.

First take a vector bundle E0 on V×S. The square-zero algebra $O\oplus E0$ defines a finite S-projective scheme Z over V×S; its closed zero section is V×S. Both Z and the zero section are flat over S. A quotient $E0\to F$ is equivalent to an ideal $I\subset E0$ inside $O_Z$; the closed subscheme W defined by I contains the zero section and has $O_W=O\oplus F$. Flatness of F is equivalent to flatness of $O_W$, since the direct summand O is S-flat. With the pullback polarization the polynomial of W is $P_O+P$. Apply the actual Hilbert supplier to Z/S and this polynomial, obtaining a projective scheme H0 and its universal ideal $I_W$.

The condition that W contain the zero section is closed, including on nonreduced tests. Indeed $I_W$ is H0-flat (kernel of a quotient of two H0-flat sheaves). For sufficiently high twist its pushforward is locally free and generates it, and the pushforward of $O_V$ of that twist is also locally free. The zero locus of the resulting bundle map represents exactly $I_W\to O_V$ being zero. Here uniform relative Serre vanishing exists because H0 is Noetherian and projective; a finite affine covering gives one sufficiently high twist. This zero locus is therefore a projective S-scheme representing Quot(E0,P) on all tests.

For general E choose a finite presentation $E1\to E0\to E\to 0$ with E0 a vector bundle. Such a presentation exists by an absolute ample twist on the projective scheme V×S. After Quot(E0,P) has been constructed, choose E1 to be a sufficiently negative sum of the same ample line bundle surjecting onto $ker(E0\to E)$. Its exponent can be made arbitrarily large. The universal quotient F is flat over this projective parameter scheme, so a sufficiently high twist has locally free pushforward and no higher cohomology. Vanishing of $E1\to F$ is consequently the zero locus of a section of a finite vector bundle (base twists on S cause no change). It cuts out a closed subscheme representing exactly quotients factoring through E. This construction uses equations of sheaf maps, so its representing bijections persist for arbitrary, including non-Noetherian and nonreduced, base changes. No boundedness theorem for stable sheaves is used.

For clarity, the base-change assertions needed here have a finite-complex proof. A coherent family flat over a Noetherian parameter base on the fixed smooth projective V has finite relative Tor amplitude, computed by a locally free relative resolution of length at most dim(V). A finite affine Cech model computes its proper pushforward and every derived base change. The pushforward is bounded coherent by F2 and has bounded Tor amplitude, hence has a finite locally free model on each parameter affine (cancel a bounded-above finite free resolution using its bounded point-Tor amplitudes). If all fiber cohomology is in degree zero, a minimal such model has only degree zero, so the pushforward is locally free and commutes with all base changes. Applying this after high twist proves the vector-bundle pushforwards in the Quot construction. Applying it to a universal family on its Noetherian parameter scheme, then pulling that universal identity back, includes arbitrary non-Noetherian tests without assuming closed-point Nakayama on such tests.

Pullback ideals. Let $f:V\to X$ be projective birational, with V a connected smooth projective threefold, X normal projective Gorenstein, f small and crepant, and every fiber at most one-dimensional. Assume f is an isomorphism outside a finite set. The preceding local duality argument proves $R{f}_{\ast }{O}_V=O_X$ and glues.

For a coherent sheaf A on X, $R^1{f}_{\ast }(f\ast A)=0$. Locally choose a finite free presentation of A, let J be the image of its pulled-back relation map into $O_V^n$, and use $0\to J\to O_V^n\to f\ast A\to 0$. The long exact sequence and cohomological dimension one give the assertion.

For torsion-free A, also $f_{\ast }f\ast A=A$. More explicitly the free-resolution/projection double complex gives the exact sequence $R^1{f}_{\ast }{\mathrm{Tor}}_1^X(O_V,A)\to A\to f_{\ast }f\ast A\to 0$. The first term is supported on the exceptional image and is therefore torsion; its map into torsion-free A is zero. The projection calculation is valid even if A has infinite projective dimension: the pushforward direction has only degrees zero and one, so each fixed total degree sees only two terms and no unbounded convergence assertion is required. In particular $R{f}_{\ast }(f\ast I_x)=I_x$ for the ideal of every point $x\in X$.

Exact quotient properness calculation. Fix a point $y\in V$ and put $P0=P_{I_y}$, computed using H. On V×X let $E=(f\times i{d}_X{)}^{\ast }{I}_{\mathrm{\Delta }}$, where $I_{\mathrm{\Delta }}$ is the universal point ideal on X×X. Form the relative projective scheme $Q=Quo{t}_{V\times X\mathrm{/}X}(E,P0)$ from §1.

At a closed point of Q there is a quotient $f\ast I_x\to F$ with polynomial P0. Its kernel K has rank zero. On the complement of $f^{-1}(x)$, the source is $O_V$; a rank-zero submodule of this line bundle on the integral V is zero. Thus K is supported on $f^{-1}(x)$. Moreover $f_{\ast }K$ injects into $f_{\ast }f\ast I_x=I_x$, so it is a torsion submodule of a torsion-free sheaf and is zero.

Euler characteristics now force the other pushforward to vanish. By §2, $\chi (f\ast I_x)=\chi (I_x)=\chi (O_X)-1$. By $R{f}_{\ast }{O}_V=O_X$ and the constant term of P0, $\chi (F)=\chi (O_V)-1=\chi (O_X)-1$. Hence $\chi (K)=0$. Since K is supported on the single fiber, $f_{\ast }K=0$, and f has cohomological dimension one, $\chi (K)=-length(R^1{f}_{\ast }K)$. Therefore $R^1{f}_{\ast }K=0$. The quotient sequence gives $R{f}_{\ast }F=I_x$, with the canonical unit map $I_x\to R{f}_{\ast }F$ an isomorphism.

This calculation is the key properness argument. It does not presume that pushforward preserves arbitrary coherent quotients, and it does not discard a torsion boundary by calling it unstable.

The universal graph-normalized kernel. First take $S=Q$ with its universal flat quotient $E_S\to F$; this is a Noetherian base. On each geometric fiber the preceding calculation gives $R{f}_{\ast }{F}_s=I_{x(s)}$. Derived base change for the S-flat proper family F therefore gives that $R{f}_{S,\ast }F$ is S-flat in degree zero. The adjunction map from the graph ideal $I_{\mathrm{\Gamma }}$ to $R{f}_{S,\ast }F$ restricts to the fiberwise unit isomorphisms in §3. Its cone is zero by the relative Nakayama criterion for bounded finitely presented flat families; consequently $R{f}_{S,\ast }F=I_{\mathrm{\Gamma }}$ canonically. All these universal complexes have finite Tor amplitude over Q and commute with derived base change, computed by finite affine Čech models. Pulling this universal isomorphism to any test $S\to Q$ therefore proves the same assertion for arbitrary non-Noetherian or nonreduced S, rather than using a closed-point Nakayama argument on an arbitrary base.

Relative crepant duality gives $f_S!O_{S\times X}=O_{S\times V}$; it is the flat base change over C of the fixed dualizing identity. Thus the inclusion $I_{\mathrm{\Gamma }}\to O_{S\times X}$ has a unique adjoint map $F\to O_{S\times V}$. Let P be its cone. It is a relative perfect complex over S: F is S-flat and coherent and the V factor is smooth of dimension three, giving a uniform finite relative locally free model. The pushforward of P is $O_{\mathrm{\Gamma }}$.

The actual heart and numerical component. Use the heart $Pe{r}^{-1}(V\mathrm{/}X)$ whose objects have cohomology only in degrees −1,0, satisfy $f_{\ast }{H}^{-1}=0$, $R^1{f}_{\ast }{H}^0=0$, and $\mathrm{Hom}(H^0,C)=0$ for coherent C with $R{f}_{\ast }C=0$. Here is the actual torsion-pair construction, rather than an assumed gluing theorem. Put $F0=\{G:f_{\ast }G=0\}$ and $T0=\{G:R^1{f}_{\ast }G=0\text{ and }\mathrm{Hom}(G,C)=0\text{ whenever }R{f}_{\ast }C=0\}$. T0 is closed under quotients and extensions by the cohomological-dimension-one long exact sequence and left exactness of Hom into C. If $T\in T0$ maps to $G\in F0$, its image J has $f_{\ast }J=0$ and $R^1{f}_{\ast }J=0$ (it is a quotient of T), so J belongs to the forbidden C category and the map is zero. For any coherent E, the sum of its T0 subobjects stabilizes by Noetherianity; it is again T0, since finite sums are quotients of extensions. Call the maximal such subobject t(E). If $f_{\ast }(E\mathrm{/}t(E))$ were nonzero, the evaluation $f\ast f_{\ast }(E\mathrm{/}t(E))\to E\mathrm{/}t(E)$ would have nonzero image. Its source belongs to T0 by §2 and the pullback–pushforward Hom adjunction; so would its image, contradicting maximality. Thus $E\mathrm{/}t(E)\in F0$, proving the torsion pair.

Its tilted aisle consists of complexes with $H^i=0$ for $i>0$ and $H^0\in T0$; its coaisle consists of complexes with $H^i=0$ for $i<-1$ and $H^{-1}\in F0$. Orthogonality follows from ordinary truncation and $\mathrm{Hom}(T0,F0)=0$. For the truncation axiom, take the homotopy fiber K of $\tau \le 0E\to H^0(E)\mathrm{/}t(H^0(E))$. Then K is in the new aisle, and its cone in E has no negative cohomology and has $H^0$ in F0, hence is in the new coaisle shifted to degree one. These constructions give a bounded t-structure. Its heart has the displayed two-term description.

The sheaf F lies in this heart: $R^1{f}_{\ast }F=0$, and every map $F\to C$ vanishes, since F is a quotient of $f\ast I_x$ and $\mathrm{Hom}(f\ast I_x,C)=\mathrm{Hom}(I_x,f_{\ast }C)=0$. Its map $F\to O_V$ is a monomorphism in this heart. This can also be checked directly: the cone P has $H^{-1}=ker(F\to O_V)$ with pushforward zero, $H^0=coker(F\to O_V)$ with $R^1$ pushforward zero, and $\mathrm{Hom}(H^0,C)=0$ because it is a quotient of $O_V$. Thus $0\to F\to O_V\to P\to 0$ is exact in $Pe{r}^{-1}$.

The full numerical point class, rather than merely one Hilbert polynomial, is obtained by selecting the union of connected components of Q on which F has the numerical class of $I_y$. This is an open and closed projective subscheme. For every vector bundle L on V, $\chi (F\otimes L)$ is locally constant in the universal flat family; it is therefore constant on each of the finitely many connected components of Q. These simultaneous values determine the numerical class. No infinite intersection of non-open loci is being taken.

Call this subscheme $Q_{pt}$. It contains V minus the exceptional locus, where the quotient is the ordinary point ideal. Conversely, a relative-perfect family of perverse quotients $O_V\to P$ of numerical point class has $R{f}_{\ast }P$ a length-one flat point family; after its canonical graph normalization, its kernel F is a flat coherent family with $R{f}_{\ast }F=I_{\mathrm{\Gamma }}$ and is a quotient of $f_S\ast I_{\mathrm{\Gamma }}$. Thus it gives the unique corresponding point of $Q_{pt}$. For infinitesimal tests this follows directly by the local flatness/Tor criterion for a bounded relative-perfect complex whose fibers are sheaves. These are exactly the families needed for the universal Kodaira–Spencer map; no assertion about arbitrary unbounded fiberwise complexes is made.

Global equivalence. For numerical perverse points P1,P2, $\mathrm{Hom}(O_V,\mathrm{\Pi })=C$ and $O_V\to \mathrm{\Pi }$ is a heart epimorphism. Any nonzero map $P1\to P2$ is therefore a heart epimorphism: its composition with $O_V$ is a nonzero scalar multiple of the fixed quotient onto P2. Its kernel K has zero numerical class and $R{f}_{\ast }K=0$. The explicit heart conditions then force $K=G[1]$ for a coherent G with $R{f}_{\ast }G=0$. Its Hilbert polynomial is $-P_G$, which cannot vanish unless $G=0$. Hence Hom(P1,P2) is C for isomorphic objects and zero otherwise.

Take the irreducible component W of $Q_{pt}$ containing the ordinary points, with its reduced structure, and restrict the universal P to W×V. W is projective of dimension three and birational to X. The point separation follows from the Quot-family correspondence in step 23.1. Here is its infinitesimal version. Put $A=\mathbb{C}[ϵ]\mathrm{/}({ϵ}^2)$ and let a tangent vector to W at w induce $P_A$. If its class in ${\mathrm{Ext}}_V^1(P_w,P_w)$ is zero, the first-order perfect-complex deformation is trivial: in finite locally free/Čech models its differential perturbation is a coboundary, removed by a change $1+ϵh$ of model. Thus $P_A\cong P_w\otimes A$, compatibly with the special fiber. Since $R{f}_{\ast }{P}_w=O_{g(w)}$, finite Čech base change gives ${\mathrm{Hom}}_{V_A}(O_{V_A},P_w\otimes A)=A$. Under this trivialization the universal quotient map $O_{V_A}\to P_A$ is a scalar multiple of the constant quotient, with nonzero residue and hence a unit scalar. Multiplying the target by its inverse makes the perverse quotient constant. The unique inverse Quot-family construction in step 23.1 then makes its point of $Q_{pt}(A)$ constant. The closed immersion $W\hookrightarrow Q_{pt}$ is injective on tangent spaces, so the original tangent vector is zero. This proves injectivity of the actual universal-family Kodaira–Spencer map without presupposing smoothness of W. Define $\mathrm{\Phi }(A)=R\pi V\ast (P\otimes L\pi W\ast A)$ and $\mathrm{\Psi }(B)=R\pi W\ast (P\vee \otimes \pi V\ast \omega V[3]\otimes L\pi V\ast B)$. The finite relative trace makes Ψ left adjoint to Φ. The convolution Q for ΨΦ is perfect on W×W by the smooth projective intermediate V. Adjunction and derived restriction give ${\mathrm{Hom}}^i(Q,k(w1,w2))={\mathrm{Hom}}^i(P_{w}1,P_{w}2)$. Thus off-diagonal point-Ext degrees are one and two by heart orthogonality and Serre duality, and support is contained in $W{\times }_X$ W, of $dimension\le 4$. The off-diagonal minimal free complex has two terms. Since it is generically exact these have equal rank; if it were nonzero, the determinant of their differential would have a codimension-one zero by the principal ideal theorem on the integral W×W. Its support instead has codimension at least two. Thus it is zero off the diagonal. For each w the diagonal restriction $E_w=\mathrm{\Psi }\mathrm{\Phi }(k_w)$ is therefore perfect and supported at w. The point-Hom identity and heart/Serre vanishing give a minimal free model in degrees [−3,0]. Under adjunction the counit ${\epsilon }_w:E_w\to k_w$ corresponds to ${\mathrm{id}}_{P_w}$, so precomposition identifies $\mathrm{Hom}(k_w,k_w)=\mathbb{C}$ with $\mathrm{Hom}(E_w,k_w)=\mathbb{C}$. In degree one precomposition is exactly the injective Kodaira–Spencer map just proved. Complete the counit to $C_w\to E_w\to k_w\to C_w[1]$. Applying $\mathrm{Hom}(-,k_w[i])$ shows $\mathrm{Hom}(C_w,k_w[i])=0$ for every $i\le 0$: at zero use the scalar isomorphism and the degree-one injection, at minus one use the scalar isomorphism, and below that use negative-Ext vanishing. Over $O_{W,w}$ a bounded-above minimal free resolution of $C_w$ consequently has no terms in degrees at least zero, so $H^j(C_w)=0$ for $j\ge 0$. The cohomology exact sequence now gives $H^0(E_w)\cong k_w$. This constructs precisely the finite-length perfect residue-field complex required by F4, with no regularity assumption on W.

Applying F4 to this exact local complex, W is smooth and the counit is an isomorphism on every point. Its kernel cone is perfect and has zero derived restriction to every point, so the transform $Db(W)\to Db(V)$ is fully faithful. In the CY3 application ${\omega }_V=O_V$. Its image is admissible; Serre duality [3] makes its left and right orthogonals agree. Hence it is an orthogonal direct summand. Db(V) is indecomposable: $O_V$ is indecomposable, every skyscraper receives a nonzero map from $O_V$, and skyscrapers detect every nonzero object by a nonzero stalk of an extremal cohomology sheaf. The image is nonzero, so the transform is essentially surjective. This constructs a genuine GLOBAL equivalence, not a gluing of arbitrary affine tilting equivalences.

Nef-transform identification. Here assume the contraction base X is terminal Gorenstein, as for an elementary small threefold flop. The preceding actual F4 application and equivalence show that W has trivial canonical line: its Serre functor is conjugate to the [3] Serre functor of V, and applying this to $O_W$ gives ${\omega }_W=O_W$. Also ${\omega }_X=O_X$, since $f\ast {\omega }_X={\omega }_V$ and $f_{\ast }{O}_V=O_X$. The pullback of a nonzero canonical form on X to W has effective exceptional zero divisor (terminal discrepancies) and no poles. This is a nonzero section of the trivial line on the connected projective W, hence has no zeros. Thus g is crepant. Terminality excludes exceptional divisors for a crepant map, so g is small. In particular its exceptional locus has codimension at least two.

The universal graph identity gives $R{f}_{\ast }\mathrm{\Phi }=R{g}_{\ast }$ by composition of the two proper correspondences and the projection formula.

First prove the exact sheaf-shift property

$C\in Coh(W)$, $R{g}_{\ast }C=0$ implies $\mathrm{\Phi }(C)[-1]\in Coh(V)$, $R{f}_{\ast }\mathrm{\Phi }(C)[-1]=0$.

We need a short regular-ring fact. If E is a perfect complex on a smooth threefold, supported in codimension at least two, and ${\mathrm{Hom}}^i(E,k(w))$ is zero outside $i=0,1,2$, then E is a sheaf. Minimal free models give amplitude [−2,0]. Its dual is a three-term free complex in [0,2]. The first kernel is torsion inside a free module and is zero. The next cohomology embeds in the cokernel of this injective first map. At a codimension-two point that cokernel has depth at least one by the depth lemma, so it has no finite-length submodule; this kills the second cohomology there. The remaining possible support is codimension three, where the same cokernel has depth at least two, again excluding a finite-length submodule. Thus the dual has cohomology only in degree two and gives a length-two free resolution of that cohomology H. Its support still has codimension at least two. In a regular local ring, the annihilator of H contains a regular sequence of length two (Cohen–Macaulayness and prime avoidance); applying the two short exact sequences for these regular elements gives ${\mathrm{Ext}}^0(H,R)={\mathrm{Ext}}^1(H,R)=0$. The length-two resolution gives no higher Ext above degree two. Hence dualizing back, $E=RHom(H,R)[2]$, is in degree zero. This uses regularity of W only after D3, not as a premise in D3.

Now suppose $A\in Db(W)$, $R{g}_{\ast }A=0$, and $B=\mathrm{\Phi }(A)[-1]$ is coherent. Then $R{f}_{\ast }B=0$ and B[1] belongs to the perverse heart. For any w, ${\mathrm{Hom}}^i(A,k(w))={\mathrm{Hom}}^i(B[1],P_w)$ vanishes outside [0,3] by the heart and CY Serre duality. Its degree-three group is dual to $\mathrm{Hom}(P_w,B[1])$, which vanishes: a map from the perverse quotient $P_w$ injects into $\mathrm{Hom}(O_V,B[1])=H^1(V,B)=0$. Hence the regular-ring fact makes A coherent.

For general coherent C with $R{g}_{\ast }C=0$ put $D=\mathrm{\Phi }(C)[-1]$. The two-column pushforward spectral sequence shows that every cohomology sheaf of any object in ker $R{f}_{\ast }$ again belongs to ker $R{f}_{\ast }$. If D is not a sheaf, its lowest negative or highest positive cohomology therefore supplies a coherent $B\in kerR{f}_{\ast }$ and a nonzero negative-degree map $B\to D$ or $D\to B$. Put $A={\mathrm{\Phi }}^{-1}(B[1])$. The preceding paragraph makes A coherent, with $R{g}_{\ast }A=0$. The negative-degree map is then a map $A\to C$ or $C\to A$ in the same negative degree, impossible for coherent sheaves. This proves the sheaf-shift property. It also proves that every coherent $B\in kerR{f}_{\ast }$ has the form Φ(A)[−1] for coherent $A\in kerR{g}_{\ast }$.

Let $D_W$ be a Cartier divisor with $-D_W$ g-nef, and let $D_V$ be its strict transform. Set $M=\mathrm{\Phi }(O_W(D_W))$; off the exceptional loci this is the corresponding line bundle, so $\mathrm{rank}(M)=1$ and $c1(M)=D_V$.

For any contracted reduced integral curve $C\subset V$ one can construct a coherent rank-one sheaf B on C with $H^0(B)=H^1(B)=0$, without assuming C rational. Choose an ample line L on C with $H^1(L)=0$. Select $h^0(L)$ distinct smooth points successively so that evaluation is an isomorphism: a nonzero remaining section cannot vanish at every smooth point of the integral C. Then $B=L(-sumofthesepoints)$ has $H^0=0$ and $\chi =0$, hence $H^1=0$. Its pushforward to V has cycle [C] and $R{f}_{\ast }B=0$.

Write $B=\mathrm{\Phi }(A)[-1]$ with A coherent and $R{g}_{\ast }A=0$. Its support is on exceptional curves, its cycle [A] is effective, and $\chi (A)=0$. The curve Euler identity proved next gives $D_V\cdot C=$ −χ(M,B) $=$ χ(M,B[1]) $=\chi (O_W(D_W),A)=-D_W\cdot [A]\ge 0$.

Thus the strict transform on V of every g-anti-nef divisor is f-nef, exactly the nef-transform characterization of the flop. This proof does not rely on a classification of contracted rational curves.

Here is the claimed curve Euler calculation. On any integral projective curve C, choose a positive line L and a sufficiently high power such that a given vector bundle $E\otimes L^n$ is generated. Generic rank-many sections are independent at the generic point and at every singular point (a finite set), giving an injection $(L^{-n}{)}^{\mathrm{rank}(E)}\to E$ whose finite-length cokernel is supported at smooth points. At those points its length is the order of the determinant section. Tensoring with a line bundle B and taking Euler characteristics consequently gives $\chi (E\otimes B)=\mathrm{rank}(E)\chi (B)+\mathrm{deg}(\det E)$; the same exact point sequences give $\chi (B\otimes L)=\chi (B)+\mathrm{deg}$ L. For an arbitrary sheaf A supported on curves, choose a generic associated-curve section of a sufficiently positive twist of its subsheaf annihilated by that curve's ideal; it embeds a negative line bundle on the integral curve and lowers the generic length. Iterating gives, modulo zero-dimensional sheaves, a finite filtration by such curve line bundles. Zero-dimensional Euler characteristic is unchanged by line-bundle twists. Hence $\chi (A\otimes O(-D))=\chi (A)-D\cdot [A]$. Apply the vector-bundle formula termwise to any finite locally free model of M: $\mathrm{rank}(M)=1$ and $c1(M)=D_V$ give $\chi (M,B)=\chi (B)-D_V\cdot C$. These are exactly the identities in the displayed nef calculation, with $\chi (A)=\chi (B)=0$. No threefold Riemann–Roch theorem is an undeclared input.