Yasufuku: GCD inequalities arising from codimension‐2 blowups

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Yu Yasufuku, "GCD inequalities arising from codimension‐2 blowups," Bulletin of the London Mathematical Society, 58(4), e70349, 2026. https://doi.org/10.1112/blms.70349

Bears on: Problem 699.

Overview

Question and principal result. The paper seeks an unconditional, weaker analogue of Silverman’s conjectural upper bound for generalized GCDs. Silverman’s Theorem 1.1, quoted on pp. 1–2, derives from Vojta’s Main Conjecture an estimate for homogeneous forms cutting out a smooth codimension-$r$ subscheme of projective space. Yasufuku treats two forms without assuming Vojta’s conjecture. In Theorem 1.2 (pp. 2–3), let $F_1,F_2\in k[X_0,\ldots,X_n]$ have degrees $d_1,d_2$, and suppose that their common zero scheme $\mathcal Y$ is a codimension-$2$ complete intersection containing none of the $n+1$ coordinate points. If $c>\max(d_1,d_2)$ and $c>n+1$ satisfies the equation in that theorem—equivalently, $f_n(c,d_1,d_2)=1$ for the function in (9), pp. 8–9—then every $\epsilon_n(d_1,d_2)>c-(n+1)$ satisfies

$$\log\operatorname{GCD}^{+}(F_1(P),F_2(P))<\epsilon_n(d_1,d_2)h(P)+\sum_{v\notin S}\lambda_v((X_0\cdots X_n=0),P)$$

outside a proper Zariski-closed subset of $\mathbb P^n$. The exceptional set may depend on $k,S,F_1,F_2$. For equal degrees $d_1=d_2=d$, the admissible threshold is the explicit expression (3), p. 3; inequalities (16)–(17), pp. 11–12, place it strictly between $d-1$ and $d$, so this improves the elementary bound (7). The numerical values stated in Theorem 1.2 include $\epsilon_3(1,2)=0.78$, $\epsilon_3(2,3)=1.69$, and $\epsilon_3(3,4)=2.64$. Example 1.3 (pp. 3–4) illustrates the result for a linear form and a coordinate form.

GCD interpretation. If $\pi:V\to\mathbb P^n$ is the blowup along $\mathcal Y$ and $E$ is its exceptional divisor, Silverman’s local-height identity (5), p. 5, gives $\lambda_v(E,P)=\min(\lambda_v(F_1=0,P),\lambda_v(F_2=0,P))$. Definitions (6), pp. 5–6, identify the sum of these local contributions with $\log\operatorname{GCD}^{+}$. For integral coefficients and primitive integral coordinates at nonarchimedean places this is the ordinary logarithmic GCD; the paper explicitly warns on p. 6 that with nonintegral coefficients or archimedean places it need not literally be a GCD. Equation (7), p. 6, gives the general comparison $h(E,P)\leq\min(d_1,d_2)h(P)$ up to a bounded function.

Method. The analytic input is the arithmetic Ru–Vojta theorem quoted as Theorem 3.2, pp. 6–7, formulated using the beta invariant of Definition 3.1 and inequality (8). The original geometric work begins with the codimension-$2$ blowup. On p. 7, hypothesis (ii) is used to prove that the pullbacks of the coordinate hyperplanes intersect properly; Cohen–Macaulayness of the blowup is invoked from cited background. The proof then computes all mixed intersections of the hyperplane class $H$ and $E$ from $(d_1H-E)(d_2H-E)=0$ (p. 7). Assuming $d_1\leq d_2$, it shows that $cH-E$ is ample for $c>d_2$ (pp. 7–8). Asymptotic Riemann–Roch applied over the ample range yields the explicit lower bound $\beta(cH-E,H)\geq f_n(c,d_1,d_2)$ in (9), pp. 8–9. Choosing $c$ with $f_n(c,d_1,d_2)>1$ and applying Theorem 3.2 to the pulled-back coordinate hyperplanes produces the asserted GCD estimate (p. 9). For $d_1=d_2=d$, Lemma 3.3 (pp. 9–11) factors the numerator-minus-denominator of $f_n$ as (10); solving the resulting quadratic gives (15), p. 11, and the nontriviality estimates (16)–(17).

Scope and limitations. The center may be reducible or nonreduced; Remark 3.4 (p. 12) explains why proper intersection and the intersection computations persist with the appropriate multiplicities. Remark 3.5 (p. 12) emphasizes that excluding the coordinate points is substantial and prevents an exceptional component from lying in two pulled-back coordinate hyperplanes. Remark 3.6 (p. 12) observes that, in equal degree, the estimate is useful chiefly for points near $S$-units. Raising both forms to powers does not strengthen the same balance: Remark 3.7 and (18), pp. 12–13, show the resulting scaling explicitly. Remark 3.8 (p. 13) gives a natural example to which the theorem does not apply because proper intersection fails. Finally, Theorem 1.4 (p. 4) proves the parallel Nevanlinna-theoretic upper bound for the simultaneous counting function of two divisors and a Zariski-dense holomorphic curve.

Relation to E699

Write the variables of E699 as $N,i,j$ to avoid confusing $N$ with the paper’s ambient dimension. Set

$$G_{N,i,j}=\gcd\!\left(\binom Ni,\binom Nj\right).$$

For every prime $p$,

$$v_p(G_{N,i,j})=\min\!\left(v_p\!\binom Ni,v_p\!\binom Nj\right).$$

Thus E699 asks whether $\sum_{p\geq i}\min(v_p\binom Ni,v_p\binom Nj)\log p>0$. A counterexample is exactly an eligible triple for which $G_{N,i,j}$ is supported entirely on the finite set of primes $p<i$. This local minimum is formally analogous to the blowup identity (5), while the sum over all places corresponds to (6).

The analogy does not yield a direct application of Theorem 1.2. For fixed $i<j$, the natural polynomial representatives

$$A_r(X,Y)=\prod_{a=0}^{r-1}(X-aY)=r!\binom{X/Y}{r}Y^r$$

satisfy $A_i\mid A_j$. Hence their common zero scheme has a codimension-$1$ component, contrary to the codimension-$2$ complete-intersection hypothesis of Theorem 1.2; moreover, the one-parameter binomial family naturally lives in $\mathbb P^1$, where the required codimension-$2$ setup is unavailable. Replacing the binomial polynomials by the integral falling factorials also introduces the factors $i!$ and $j!$, which can alter the exact assertion that a common prime is at least $i$. Keeping rational coefficients avoids that scaling but invokes the paper’s warning after (6) that $\operatorname{GCD}^{+}$ need not be the literal integer GCD.

One could try to encode the values using additional projective coordinates and different forms, but the paper supplies no such construction. Even then, its estimate holds only outside a form-dependent proper Zariski-closed set; the one-parameter curve representing varying $N$ could lie in that set. The forms and their degrees would also vary with $i,j$, whereas E699 requires a uniform assertion over all eligible triples.

Conceptually, (5)–(7) provide a useful geometric language for the total common valuation, and Theorem 1.2 could enter a contradiction argument if an admissible encoding, avoidance of the exceptional set, and an independent lower bound for $G_{N,i,j}$ were available. But the theorem is an upper bound for the total GCD, not a lower bound and not a prime-support theorem; varying $S$ does not isolate or force a positive contribution from primes $p\geq i$. It therefore neither proves E699 nor excludes its counterexamples. Its relevance is methodological and currently weak: it explains how simultaneous divisibility can be represented by an exceptional divisor, while the decisive large-prime step required by E699 is absent.