The generalized Mukai conjecture

1. From a lower bound and independence to the inequality

Proposition 3.1 alone does not give independence. Combine it with the recurrence in Lemma 2.1 and the density of free degrees in Lemma 3.3. If every joint eigenvalue tuple were algebraically dependent, the resulting superexponential upper bound would contradict the descendant lower bound. This proves Proposition 4.1. The details are separated into the sections on the lower bound and independence.

Proposition 3.1 · p. 6 · Lemma 2.1; (4)–(5) · p. 5 · Lemma 3.3 · pp. 10–11; Proposition 4.1 (proof) · p. 12 · Proposition 4.1; (8)–(11) · pp. 11–13

Apply Proposition 5.3 to the independent tuple. It yields linearly independent curve degrees $\gamma_1,\ldots,\gamma_r$ with $S_{\gamma_1}\cdots S_{\gamma_r}\ne0$. Writing $d_\gamma=-K_X\cdot\gamma$, one obtains

The first inequality uses $d_{\gamma_j}\ge\iota$ for positive stable-map degrees; the second counts the grading drop of the nonzero product. This proves the inequality in the main theorem. The alternating-trace section explains how to extract this product.

Proposition 5.3; (15) · p. 15

2. Equality forces the product of projective spaces

If $r(\iota-1)=n$, all the preceding bounds are equalities and each $d_{\gamma_j}=\iota$. The independent nonzero product from Proposition 5.3 is exactly the additional input required by Proposition 6.4. Lemma 6.3 passes to actual proper rational-curve families. The corrected chain-locus dimension bound and cyclic rotation make every family covering. The product characterization gives

The equality section explains the hypotheses and applications of these external results.

Lemma 6.3 · pp. 16–17 · Proposition 6.4 · p. 17 · Mukai chain bound [BCDD] · Theorem 5.2 · print p. 623 / PDF p. 23 · Mukai chain bound [ERR] · Erratum · p. 1 · Product characterization [OC] · Theorem 1.1 · print p. 271 / PDF p. 2

3. The product attains equality

Conversely, let $X=(\mathbb P^{\iota-1})^r$ with $\iota\ge2$. Its dimension is $r(\iota-1)$ and its Picard number is $r$. If $H_j$ is the pullback of the hyperplane class from the $j$th factor, then $-K_X=\iota\sum_jH_j$. A line in one factor has anticanonical degree $\iota$. Every irreducible rational curve has nonnegative integral intersections with the $H_j$, at least one of which is positive. Thus the pseudoindex is $\iota$, and equality holds. This completes both directions of Theorem 1.1.

Theorem 6.2 · p. 16; Proposition 6.4; converse · p. 17

1. A contraction preserving the unobstructed open part

Let $\beta$ be free, and set $d=d_\beta$, $b=ad$, and $h=d-2$. For a very general point $x$, put $F=\mathrm{ev}^{-1}(x)\subset\overline M_{0,1}(X,\beta)$. The open substack $U$ with irreducible source is nonempty and smooth, with $[F]^{\mathrm{vir}}|_U=[U]$. The manuscript retains the marked component and replaces the other trees by common zeros at their attachment points. In §3.2, $L^+=L(\sum_e e\Delta_e)$ transfers all degree to the marked component without changing a neighborhood of the marking. Normalized coefficients define $\Phi:F\to W$ into a weighted projective stack $W$, with $\psi=\Phi^*\xi$, $\xi=c_1(\mathcal O_W(1))$, and generic stack degree one on $U$.

§§3.1–3.3 · pp. 6–8 · Intermediate moduli [MM] · Definition 1.2; Proposition 1.3 · pp. 5–6; Lemma 3.3 (proof) · pp. 17–18 · Chow ring [MMC] · Proposition 1.7 · p. 5

2. It is the boundary pushforward that vanishes

Let $d_0$ be the anticanonical degree of the marked component and $u$ the number of attached trees. The retained data have dimension at most $d_0+u-2$. Each tree has degree at least $\iota\ge2$, so $d-d_0\ge2u$ yields

The boundary term in Chow localization therefore vanishes after pushforward. Thus $\Phi_*[F]^{\mathrm{vir}}$ is a nonempty effective $h$-cycle consisting of closures of components of $\Phi(U)$. The argument does not assume that $[F]^{\mathrm{vir}}$ itself is effective or that all tail maps are unobstructed.

§3.4 · pp. 8–9

3. Turning stack degrees into a numerical bound

All weights of $W$ are at most $b$. Lemma 3.2 gives $\int_Z\xi^h\ge b^{-(h+1)}$ for every integral $h$-dimensional closed substack $Z\subset W$. A monomial degeneration of the weighted homogeneous ideal reduces the estimate to coordinate substacks, of degree $1/(w_0\cdots w_h)$. The projection formula gives $\int_{[F]^{\mathrm{vir}}}\psi^h=\int_{\Phi_*[F]^{\mathrm{vir}}}\xi^h$, hence the desired lower bound. Stabilizer denominators are essential to this numerical estimate.

Lemma 3.2; Proposition 3.1 (proof) · pp. 9–10

1. A recurrence with a separate degree-zero initial value

Set $v_0=\mathrm{pt}$ and, in positive degree, $(v_\beta,b)=\sum_{\ell\ge0}\langle\tau_\ell(\mathrm{pt}),b\rangle_\beta$. Lemma 2.1 gives

The divisor correction vanishes because $D_j\cup\mathrm{pt}=0$. In topological recursion, the degree-zero three-point side contributes classical cup product, while the primary term contributes $A_{j,\beta}v_0$. The initial value $v_0$ must not be identified with an unstable degree-zero two-point invariant. The second identity uses the string equation. The cited source explicitly states that this equation remains valid for nonconvex smooth projective targets using virtual classes.

Lemma 2.1; (4)–(5) · p. 5 · Correlator relations [KM] · (4a) · print p. 388 / PDF p. 4; Lemma 1.4, (12) · print p. 391 / PDF p. 7 · Rational curves on hypersurfaces [PA] · §1.2 · print p. 311 / PDF p. 6

2. Finite shifts and factorial normalization

Set $w_{\beta,k}=(d_\beta+k)!v_{\beta,k}$. Then $(A_j(E)w)_{\beta,k}=\beta_jw_{\beta,k}/(d_\beta+k)$. Combining these identities with the coefficients of $-K_X$ and inverting the grading-raising classical term by a finite nilpotent series gives $\|w_\beta\|\le B^{d_\beta+1}$. The positive shifts here form a finite set and decrease anticanonical degree. The shifts arising from the polynomial relation below need not have this monotonicity.

§4.1; (6)–(7) · pp. 11–12

3. A free direction avoiding a relation

Suppose every joint diagonal tuple over $L=\overline{\mathbb C(q_1,\ldots,q_r)}$ is algebraically dependent. Multiplying finitely many relations, then using simultaneous triangularization and homogeneity (3), gives a nonzero homogeneous polynomial $Q$ with $Q(A_1,\ldots,A_r)=0$. Lemma 3.3 supplies a free degree $\eta$ with $Q(\eta)\ne0$. That lemma uses rational connectedness of Fano manifolds, Tian–Zong's generation of $\mathrm{CH}_1$ by rational curves, and comb smoothing. Taking handle $\mathbb P^1$ and a twist of degree $-1$ gives $H^1(f^*T_X(-1))=0$ after smoothing, hence freeness. Every rational curve degree is a difference of free degrees; closure under addition then gives Zariski density.

Lemma 3.3 · pp. 10–11; Proposition 4.1 (proof) · p. 12 · One-cycles [TZ] · Theorem 1.3 · p. 2; Proposition 2.4 · p. 4 · Fano rational connectedness [CA] · Corollary 3.2 · print p. 543 / PDF p. 6 · Rational curves [DE] · Definition 2.20 · print p. 29 / PDF p. 30; Theorem 2.49 · print p. 43 / PDF p. 44

4. Linearly many iterations force incompatible decay

Polynomial differencing with $p=\beta/d_\beta$ held fixed gives $\|w_\beta\|\le(C/d_\beta)\max_{\delta\in U}\|w_{\beta-\delta}\|$ near $\eta/d_\eta$. For $\beta=s\eta$, choose $\epsilon>0$ so that the degree and direction remain controlled through $\lfloor\epsilon d_\beta\rfloor$ finite shifts. Using (7) after these iterations yields

On the other hand, $s\eta$ remains free, and Proposition 3.1 gives

The logarithm of the upper bound is $-\epsilon d_\beta\log d_\beta+O(d_\beta)$, whereas the lower bound is $-O(d_\beta)$. Their incompatibility proves the algebraic independence asserted in Proposition 4.1.

Proposition 4.1; (8)–(11) · pp. 11–13

1. Alternation removes derivatives of the basis

When $A_0$ commutes with every $A_j$, the manuscript proves simultaneous conjugation invariance of

A change of basis $G(q)$ introduces $K=G^{-1}dG$. Setting $B_j(t)=dA_j+t[K,A_j]$ gives $[A_j,B_\ell(t)]=[A_\ell,B_j(t)]$. Pairing permutations with exchanged labels cancels the commutator terms in the differentiated trace. The proof does not differentiate $A_0$ or require $A_0$ to commute with $dA_j$.

Lemma 5.1; (12)–(14) · pp. 13–14

2. Selecting one joint tuple

Choose an interpolation polynomial $p$ taking value one at the independent tuple $\lambda$ and zero at the other distinct joint tuples, and set $A_0=p(A_1,\ldots,A_r)$. In a common upper triangular basis,

The matrix $A_0$ need not be an idempotent projector on a generalized eigenspace: selecting the diagonal entries suffices. This is another point at which the argument avoids semisimplicity.

Lemma 5.2 · pp. 14–15

3. Returning to the fixed basis and counting degrees

In the fixed basis, $dA_j=\sum_{\gamma>0}q^\gamma\gamma_jS_\gamma\omega_\gamma$, where $\omega_\gamma=\sum_\ell\gamma_\ell\,dq_\ell/q_\ell$. A nonzero summand in the expansion of $\Theta$ has both $\omega_{\gamma_1}\wedge\cdots\wedge\omega_{\gamma_r}\ne0$ and $\operatorname{Tr}(A_0S_{\gamma_1}\cdots S_{\gamma_r})\ne0$. The first gives linear independence of degree vectors; the second gives a nonzero ordered operator product. Each correspondence lowers codimension by $d_{\gamma_j}-1$. Since the available degrees are $H_0,\ldots,H_n$,

This is Proposition 5.3 and proves the inequality in Theorem 1.1.

Proposition 5.3; (15) · p. 15

1. From supports of correspondences to a nonempty chain

The two-evaluation image $Z_\gamma\subset X\times X$ contains the support of the correspondence inducing $S_\gamma$. If the incidence fiber product matching consecutive evaluations were empty, the composition would vanish. A nonzero product therefore supplies a chain. Order the curves according to composition, reversing and relabelling the degree list if necessary. Total degree $\iota$ forces each stable map to have exactly one nonconstant component, birational onto its image. Take the full irreducible components of $\mathrm{RatCurves}^n(X)$ containing these curves. After forgetting the markings, a stable limit has neither splitting, multiple covers, nor contracted trees. These components are consequently proper.

Lemma 6.3 · pp. 16–17

2. The ordered-locus bound makes every family covering

Theorem 5.2 of the Mukai chain bound gives dimension at least $\sum_j(-K_X\cdot V^j-1)$ for a nonempty ordered endpoint locus of numerically independent proper irreducible components. The erratum requires irreducible components of $\mathrm{RatCurves}^n(X)$; arbitrary proper subfamilies do not suffice. Lemma 6.3 provides precisely this condition.

At equality the lower bound is $n$, so the last family $V^r$ is covering. Properness makes its evaluation image closed, hence all of $X$. Prepending a member of $V^r$ to the original chain and dropping the last member rotates the order. Repeating this operation puts each $V^j$ last in a nonempty chain. The same dimension bound makes every family covering.

Theorem 6.1 · p. 16; Proposition 6.4 · p. 17 · Mukai chain bound [BCDD] · Theorem 5.2 · print p. 623 / PDF p. 23 · Mukai chain bound [ERR] · Erratum · p. 1

3. Checking Occhetta's hypotheses

The families are unsplit, covering, numerically independent, and all have anticanonical degree $\iota$. With $n_j=\iota-1>0$, one has $\sum_j n_j=n$, so Occhetta's Theorem 1.1 gives $X\simeq(\mathbb P^{\iota-1})^r$. Conversely, computing the dimension, Picard number, and pseudoindex of this product proves equality.

Theorem 6.2 · p. 16; Proposition 6.4; converse · p. 17 · Product characterization [OC] · Theorem 1.1 · print p. 271 / PDF p. 2