The closed vertical Brill--Noether interval and its Gamma boundary

We prove the precise strict-BG consequence used below. Suppose a positive-rank first-heart object E is semistable at (0,w), $w>0$, and $0<\mu (E)=D\mathrm{/}r\le \epsilon$ but C₂(E)≥−D/2. Choose such a counterexample with lexicographically least (Δ,D), where Δ≥0 is an integer by the actual tilt-discriminant supplier and $D>0$ is an integer. At an equal-finite-phase split $E=A+B$, both denominators $D_i$ are positive and add to D. The weighted identity $r\mathrm{/}D=\sum (D_i\mathrm{/}D)(r_i\mathrm{/}{D}_i)$ selects a factor with $r_i\mathrm{/}{D}_i\ge r\mathrm{/}D>0$, so $0<{\mu }_i\le \mu$. Its numerical line is $C_{2,i}=\nu D_i+w{r}_i$. Since $w>0$ and ${\mu }_i\le \mu ,C_{2,i}\mathrm{/}{D}_i=\nu +w\mathrm{/}{\mu }_i\ge \nu +w\mathrm{/}\mu =C_2\mathrm{/}D\ge -1\mathrm{/}2$. It is another counterexample. At this fixed ray put $a=2w$ and $\mathrm{\ell }=\nu$. The identity $\mathrm{\Delta }=(1+{\mathrm{\ell }}^2\mathrm{/}a)D^2-a(r+\mathrm{\ell }D\mathrm{/}a{)}^2$ identifies the $\mathrm{\Delta }\ge 0,D>0$ cone with a forward two-dimensional Lorentz cone. Its bilinear polarization satisfies ⟨x,y⟩≥sqrt(Δ(x)Δ(y)), with equality only for proportional nonzero vectors unless one is null; a timelike vector pairs strictly positively with every nonzero future vector. Therefore each nonzero factor has strictly smaller Δ than a positive-Δ sum. If the total Δ is zero, both factors are proportional null vectors, and the selected factor has smaller positive D. Either contradicts the minimum, so this minimum has no proper equal-finite-phase split. Its semistable parameter set in $w>0$ is closed by its fixed first-heart subobject inequalities. It is also open: the uniform tilt-discriminant support bound on each compact positive w interval and the old/new slope inequalities bound the numerical triples of possible nearby first destabilizing HN factors, exactly as in the discriminant supplier. There are finitely many such triples; loss of semistability would force an equal-finite-phase split at one of their equality parameters. Null subobjects are absent independently of w; equal-denominator subobjects have null quotients and cannot destabilize at any w. Thus the minimum is semistable for every $w>0$. The actual large-parameter classification makes this positive-rank object an ordinary μ-semistable torsion-free sheaf. Take its finite ordinary stable-factor filtration, all of slope μ, and choose a factor J with C₂(J)/r(J) at least the rank-weighted average C₂(E)/r(E). Then C₂(J)≥−D(J)/2 and Δ(J)≤(r(J)/r(E))²Δ(E). A proper rank fraction lowers Δ when $\mathrm{\Delta }(E)>0$, or lowers D when $\mathrm{\Delta }(E)=0$. If there is just one factor, E itself is stable. In every case minimality yields an ordinary μ-stable counterexample with $0<\mu \le \epsilon$, contradicting the stated strict BG₃(ε) hypothesis. Consequently every positive-rank tilt-semistable object at (0,w) in that small slope interval satisfies $C_2<-D\mathrm{/}2$. This establishes the implication with all rank selection and strict descent inequalities, without a full BMT or flatness input.

Consequently a NEGATIVE-rank G semistable at (0,w), with $C_1(G)>0$ and ${\nu }_{\mathrm{B}\mathrm{N}}(G)\le 1\mathrm{/}2$, has $\mu (G)<-\epsilon$. Indeed if $-\epsilon \le \mu (G)<0$, its actual truncated finite-part dual has positive slope −μ(G)≤ε, first coordinates (−C₀,C₁,−C₂), and is semistable at the same parameter. The preceding strict inequality gives ${\nu }_{\mathrm{B}\mathrm{N}}(G)>1\mathrm{/}2$, contradiction. Hence rk(G)≥−C₁(G)/(vε); for nonnegative rank the same bound is automatic.

For rational $b>0,O_X[1]$ is weak-tilt-stable at every $w>b^2\mathrm{/}2$. One direct proof avoids any classification of flat bundles. If unstable, its last HN quotient is $Q=G[1]$, with a coherent sequence $0\to F\to O_X\to G\to T\to 0,G$ torsion-free in $F_b$ and $T\in T_b$. The map $O_X\to G$ is nonzero (otherwise $G=T$ would lie in both parts of the torsion pair), hence injective. Positive rank or divisorial degree of T makes $\mu (G)>0$; if $\mu (G)=0$ then T has codimension≥2 and rank $G=1$, so reflexivity of $O_X$ gives $O_X\subset G\subset G\ast \ast =O_X$ and Q is not proper. A proper finite-phase quotient thus has $0<\mu (G)<b$. Put $a=2w-b^2$ and $y=C_0(G)\mathrm{/}{d}_b(Q)>1\mathrm{/}b$. Tilt Δ(Q)≥0 gives ${\lambda }_a(Q)\ge -1\mathrm{/}(2y)+ay\mathrm{/}2>-b\mathrm{/}2+a\mathrm{/}(2b)={\lambda }_a(O_X[1])$, contradicting a smaller-slope last quotient. The same inequality excludes proper equal-finite-phase quotients, proving strict weak stability; null quotients have infinite slope and do not obstruct it.

At such a point write $S=O_X[1]$. A same-finite-phase semistable subobject $A\subset S^h$ is $S^k$. For a nonzero coordinate projection A→S, its image has phase at least that of A and at most that of S, hence the same phase. Stability of S makes its cokernel zero or numerically zero; the latter is zero-dimensional and impossible because $\mathrm{Hom}(S,T)={\mathrm{Ext}}^{-1}(O_X,T)=0$ for a coherent zero-dimensional T. Thus the projection is onto S. Its kernel is a same-phase subobject of $S^{h-1}$; induct. The extension splits because its pushout along the kernel's embedding in $S^{h-1}$ is the split ambient extension, and that embedding is a constant matrix with a constant left inverse (End $S=H^0(O_X)=C$). Hence $A=S^k$. No assertion that every numerically flat bundle is trivial is used.

Let E have positive C₁ and be vertically BN-semistable: E∈B₀ and semistable for every $0<w<w_0$. Then $\mathrm{Hom}(S,E)=0$ because S∈C₀ and E has finite slope there. For $W=\mathrm{Hom}(O_X,E)$, the evaluation cone C fits 0→E→C→S⊗W→0 in B₀ and, whenever E belongs to $B_b$, also in $B_b$. Applying Hom(S,−) to the evaluation triangle shows $\mathrm{Hom}(S,C)=0$: its preceding term Hom(S,E) is zero and its next map is the injective evaluation inclusion $W\to \mathrm{Hom}(O_X,E)$.

At a $B_b$ point where E is semistable with slope ${\nu }_E<{\nu }_S=s$, the cone's finite HN factors have slopes in $[{\nu }_E,s]$. Its upper endpoint is STRICT. If a highest semistable factor A has slope s, take its image I in $S^h$. Semistability gives ν(I)≥s from its quotient of A and ν(I)≤s from its subobject of $S^h$; I cannot be null, since $S^h$ is finite-slope semistable. Thus I has phase s. The kernel K is zero or a same-phase semistable positive-denominator object: a null kernel is excluded by finite-slope semistability, and charge additivity handles the positive case. Its inclusion into E is impossible by strict phase orthogonality. Therefore A embeds into $S^h$ and is $S^k$ by the preceding paragraph, contradicting $\mathrm{Hom}(S,C)=0$. This supplies the actual kernel argument behind the strict upper slope, not the invalid inference that $\mathrm{Hom}(A,E)=0$ alone makes $A\to S^h$ injective.

Put $D=C_1(E)>0,r=C_0(E),\nu =C_2(E)\mathrm{/}D\in [-1\mathrm{/}2,1\mathrm{/}2]$. Choose a positive rational $\eta <\min (1\mathrm{/}4,1\mathrm{/}(8D))$ and set $s=\max (\nu ,0)+\eta$. Choose rational $b>0$ sufficiently small and $w=bs$. The line of E through (b,w) crosses $b=0$ at $c_E=bD(s-\nu )\mathrm{/}(D-br)>0$,

which is less than w₀ for small b. Its two endpoints $(0,c_E)$ and (b,bs) are above the parabola and belong to the same open line segment. Exact numerical-line transport therefore makes the SAME E semistable at (b,bs), without assuming full-neighborhood BN semistability. Its slope is ${\nu }_E=(\nu D-bsr)\mathrm{/}(D-br)$, and $s-{\nu }_E=D(s-\nu )\mathrm{/}(D-br)>0$.

Take the cone HN filtration there. Each factor $G_i$ has ${\nu }_E\le {\nu }_i<s$ and positive $d_b$. Its numerical line has strictly positive intercept $c_i=b(s-{\nu }_i)$, bounded by bD(s−ν)/(D−br), which tends to zero as b does. Both 0 and b are in this line's upper-parabola interval, so exact line transport makes $G_i$ an actual semistable object in B₀ at $w_i=c_i>0$, with $D_i=C_1(G_i)>0$. All original HN triangles become triangles of B₀ objects by extension closure, hence a finite filtration of the SAME cone there. Its positive integers $D_i$ sum to D.

Write $q_i=C_2(G_i)\mathrm{/}{D}_i$ and $x_i=C_0(G_i)\mathrm{/}{D}_i$. The line equation gives $q_i={\nu }_i+c_i{x}_i$. If $x_i\le 0,q_i<s$. If $x_i>0$ and $q_i>0,\mathrm{\Delta }(G_i)\ge 0$ gives $2x_i{q}_i\le 1$ and therefore $q_i\le [{\nu }_i+\sqrt{{\nu }_i^2+2c_i}]\mathrm{/}2$ ≤[s+sqrt(s²+2 max $c_i)]\mathrm{/}2<s+\eta$

for sufficiently small fixed b. Nonpositive $q_i$ also satisfy this upper bound. Hence every factor has $q_i<\max (\nu ,0)+2\eta \le 1\mathrm{/}2+2\eta$. But $2C_2(G_i)$ is integral, and $1\le D_i\le D$. If $q_i>1\mathrm{/}2$, then $q_i-1\mathrm{/}2\ge 1\mathrm{/}(2D_i)\ge 1\mathrm{/}(2D)$, contradicting $2\eta <1\mathrm{/}(4D)$. Thus EVERY factor satisfies $q_i\le 1\mathrm{/}2$, including when $\nu (E)=1\mathrm{/}2$. This is a finite argument at one chosen b; there is no limiting HN filtration or exchange of a limit and a sum.

Section 1 bounds every factor's rank by $-D_i\mathrm{/}(v\epsilon )$. Summing the actual finite filtration gives h⁰(E)≤rk(E)+D/(vε).

The argument applies to vertically BN-semistable E of EITHER sign ν, throughout the closed interval [−1/2,1/2]. The intercept construction, not an asserted BN-stability of the cone, supplies the needed positive-parameter factors.

Assume $K_X$ is numerically trivial. Apply the same result to $G=\tau \le 0\mathrm{RHom}(E,K_X)[1]$. The actual finite-part duality gives $C_0(G)=-C_0(E),C_1(G)=D,C_2(G)=-C_2(E)$ and vertical semistability at all sufficiently small w. Tensor by the numerical canonical line preserves the heart and numerical slope. Its zero-dimensional defect triangle $G\to \mathrm{RHom}(E,K_X)[1]\to T[-1]$ makes $\mathrm{Hom}(O_X,G)=\mathrm{Hom}(O_X,\mathrm{RHom}(E,K_X)[1])$, because the two relevant negative Hom groups of T vanish. The locally-free smooth Serre-duality supplier extends here to the required bounded coherent complex without an extra theorem premise: on smooth projective X take a bounded locally-free resolution, apply the supplied trace pairings termwise, and totalize their finite cohomology complexes. Naturality with respect to the differentials gives the perfect pairing of the two total complexes; its cohomology pairs degree 1 of $\mathrm{RHom}(E,K_X)$, equivalently degree zero of $\mathrm{RHom}(E,K_X)[1]$, with degree 2 of RΓ(E), since dim $X=3$. Thus it identifies this Hom space with ${\mathrm{Ext}}^2(O_X,E)\ast$. Hence h²(E)≤−rk(E)+D/(vε).

Explicitly, hypercohomology $h^{-1}(E)=\dim \mathrm{Hom}(O_X[1],E)=0$ and $H^3(E)\ast =\mathrm{Hom}(E,K_X)=0$ by negative heart Hom to $K_X[1]\in B_0$; all other cohomology outside degrees0,1,2 vanishes. Thus χ(E)≤h⁰(E)+h²(E)≤2D/(vε). This use of BOTH rank bounds is why full-neighborhood BN stability and its below-O-line Ext² vanishing are unnecessary.

Let $t=t{d}_2(X)H\mathrm{/}v$, choose rational γ≥4/(vε)+|t|, and put $\mathrm{\Gamma }=\gamma H^2-t{d}_2(X),g=\mathrm{\Gamma }H\mathrm{/}v=\gamma -t\ge 0$. RR gives $\chi (E)=c{h}_3(E)+t{d}_2(X)c{h}_1(E)$, so $U=c{h}_3(E)-\mathrm{\Gamma }c{h}_1(E)\le (2\mathrm{/}(v\epsilon )-\gamma )D$. The CORRECT boundary quadratic form includes its Γ-rank term: $Q_{0,0}^{\mathrm{\Gamma }}\mathrm{/}2=2C_2^2-3g$ C₀C₂−3DU.

Tilt Δ≥0 gives C₀C₂≤D²/2 for either sign. Consequently $Q_{0,0}^{\mathrm{\Gamma }}\mathrm{/}2\ge 2C_2^2+(3\mathrm{/}2)(\gamma +t-4\mathrm{/}(v\epsilon ))D^2\ge 0$.

This proves the boundary for ALL vertically BN-semistable E with positive D and $\mathrm{\mid }{\nu }_{\mathrm{B}\mathrm{N}}\mathrm{\mid }\le 1\mathrm{/}2$. It supersedes the weaker full-neighborhood and zero-only boundaries for the global propagation. No projectively-flat equality theorem is needed.