On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks

Nathan Grieve

REVIEW

On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks

Nathan Grieve

Number Theory · Submitted 28 Sep 2026 · 26 references

Pith's verdict: A genuine conditional-theory paper with an honest lineage, but the hard direction of the equivalence rests on Lemma 3.7's condition (3.2), which is not satisfiable for dim >= 2 as written — the fix is to excise Supp(D*) — and the stack theorems have fixable statement gaps.

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arXiv ID2609.36300 (version 1)

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CitationNathan Grieve. "On Vojta's harder implication, height inequalities for admissible pairs, points of bounded degree and Deligne-Mumford stacks." arXiv:2609.36300 (2026). https://pith.science/paper/RPMDB2NL

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Editor's note

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Colleague — here's the short read. What's actually new: Conjectures 1.2–1.3 cast Vojta's bounded-degree inequality in Levin's admissible-pairs language, Theorem 1.4 proves the logical equivalence of the three forms (adapting Vojta's harder-implication machinery on [26, pp. 1115–1116]), and Theorems 1.6–1.7 with Theorem 5.3 transfer the conditional non-Zariski-density statement to Deligne-Mumford stacks with projective coarse space, resting on Abramovich–Várilly-Alvarado's slicing theorem. The author is transparent about following Vojta's template, and Conjecture 1.1 implies 1.2 via the D0 = 0 specialization, which is legitimate. Theorem 5.3's algebra with Silverman's discriminant bound checks out; the m-factors work out and the positivity condition n(1−ε) > mm0(2r−2) is right. The soft spot that matters is Lemma 3.7's assumption (3.2), and the stress-test note is correct for dim >= 2. For x in Supp(D*)\Supp(D1 ∪ D0), the cover ramifies at x' but x' avoids Supp(D'), so n_S^(1)(D',x') = 0 identically, while the discriminant difference is strictly positive whenever the relevant ramification place sits outside S. The bad points fill a divisor, so no fixed finite S absorbs them, and Remark 3.8's one-sentence appeal to Hermite's theorem does not address this. (In dim 1 the bad set is finite, so the problem is milder.) This inequality is precisely what converts Conjecture 1.2's counting data into the discriminant term, so the hard direction of Theorem 1.4 is unproved as written. It is repairable — excise Supp(D*) into the exceptional Z and prove Lemma 3.7 only for x outside Supp(D*) — but the repair has to be written down, and Remark 3.8 needs replacing. Two smaller items: Theorem 1.6's displayed inequality lives on Y rather than X, so the statement does not literally deliver its claimed conclusion on the stack; and Theorem 1.7 needs π chosen as in Theorem 1.5(ii) for the normal-crossings hypothesis, which the statement omits. Both look fixable. The abstract also runs ahead of the conditionality: the stack Bombieri–Lang conclusion is conditional on Conjecture 1.1, which the body states and the abstract does not. Net: the skeleton is credible, the lineage is honest, and most flaws are repair cracks rather than load-bearing failures. The one exception is condition (3.2) — that gap must be closed before the equivalence is solid. It deserves a serious referee; I'd send it out for a conditional-looking revision.

Referee report

Significance

If the equivalence Theorem 1.4 can be repaired, the paper makes a useful contribution: it clarifies the relationship among Vojta's classical bounded-degree inequality, Levin's admissible-pairs formulation, and the truncated form, in the spirit of Vojta's abc-analysis. The stack application is a clean and honest conditional reduction: it quotes the published covering theorem of Abramovich and Varilly-Alvarado, uses the cited Silverman discriminant bound, and the algebraic steps in the proof of Theorem 5.3 (equations (5.5)-(5.8)) can be verified by hand and are consistent with the hypothesis (5.3). The paper openly acknowledges its limitations: the local-function framework is restricted to divisors defined over the base field (Section 2.2), and Examples 5.5 and 5.6 concede that the curve applications do not improve known results. The main reservation is that the hard direction of Theorem 1.4 rests on a hypothesis, condition (3.2) of Lemma 3.7, that is not satisfiable in the precise way the proof applies it; the intended repair (excising the auxiliary divisor from the admissible points) is local and does not change the paper's conclusions, but it is not in the manuscript.

Major objections (3)

[Section 3, Lemma 3.7 (condition (3.2)) and Remark 3.8] Condition (3.2) is not satisfiable in the application made in the proof of Theorem 1.4. By Lemma 3.6(iii), the cover π is unramified only outside π^{-1}(Supp D*), where D* ≥ D1 is the auxiliary divisor constructed in Lemma 3.6. Take a bounded-degree point x ∈ Supp(D*)\Supp(D0+D1); such points exist generically on components of D* not contained in D0∪D1. Then x is in the domain of (3.1) (since x ∉ Supp(D0+D1)), but every lift x' ∈ π^{-1}(x) lies over the ramification locus, so d_{K,S}(x') - d_{K,S}(x) > 0 whenever any ramified prime is outside S. At the same time x' ∉ Supp(D0'+D1') (as x ∉ Supp(D0)∪Supp(D1)), so n_S^{(1)}(D',x') = 0, and (3.2) demands a positive quantity be ≤ 0. The one-sentence justification in Remark 3.8 does not address this obstruction: enlarging S moves ramified primes out of the left-hand side but cannot raise the identically zero right-hand side, and since the family of such x is infinite, no finite enlarged S can absorb all their ramified primes. The hypothesis of Lemma 3.7 is therefore logically inconsistent with the construction from Lemma 3.6 used in its application.
[Section 4, proof of Theorem 1.4] The proof text states that π is 'unramified outside of π^{-1}(Supp D1)', which contradicts Lemma 3.6(iii), where π is unramified only outside π^{-1}(Supp D*). This conflation of D1 with the larger divisor D* is precisely the source of the failure of (3.2) described above, and it is load-bearing: the argument of Lemma 3.7 is exactly the step that converts the discriminant jump of the ramified cover into the truncated counting term needed to derive Conjecture 1.3 from Conjecture 1.2. The repair is local: restrict the admissible points in (3.1) to x ∉ Supp(D*), and enlarge the exceptional set Z to contain Supp(D*). For x ∉ Supp(D*), π is unramified over x, so d_{K,S}(x') = d_{K,S}(x) (no ramified primes, by (2.3)), and x ∉ Supp(D1) gives n_S^{(1)}(D1,x) = 0 and n_S(D1',x') = 0, so the conclusion of Lemma 3.7 holds with both sides trivial and (3.2) is satisfied. The final inequality (1.7) then holds outside the proper closed set Z ∪ Supp(D*). As written, without this excision, the hard direction of Theorem 1.4 is unproved.
[Section 1, Theorem 1.6] The proof of Theorem 1.6 is a single sentence and does not demonstrate the descent from Y to X. Passing from Conjecture 1.3 on Y to Conjecture 1.3 on X requires explicit comparisons: the truncated counting function n_S^{(1)} on Y is normalized using [K(y'):Q] while on X it is normalized using [K(x):Q] (equation (2.5) with the |·|_w conventions of Section 2.1), so one must justify a lower bound for n_S^{(1)}(D1,x) in terms of n_S^{(1)}(D1',y'); one must also use d_K(y') ≥ d_K(x) and the canonical-bundle inequality of Lemma 3.1 to control the discriminant difference. These steps are not given. Since Theorem 1.7 is deduced through Theorem 5.3 rather than through Theorem 1.6, this gap does not break the main application, but Theorem 1.6 as stated needs a real proof or a restatement as a remark.

Minor objections (5)

[Section 3, Lemma 3.1] Lemma 3.1 contains two typos: the zero locus of the section should be Supp(D') ∪ Ram(π) (the text reads 'Supp(D') S Ram(π)'), and the phrase 'that that vanishes only' should be 'that vanishes only'.
[Section 3, Lemma 3.7] In the statement of Lemma 3.7, the symbol D in 'x ∈ X(K̄)\Supp D' of (3.1) is not defined within the lemma; it should read Supp(D0) ∪ Supp(D1).
[Section 4, proof of Theorem 1.4] The condition on ϵ' is stated as 'choose ϵ' > 0 so that ϵ' < ϵc/e'. As written, this does not imply the needed inequality c/e + ϵ' ≤ ϵ used to absorb (1/e)h_{O(D1)}(x) + ϵ'h_{L'}(x') into ϵh_L(x) (take c = e = 1 and ϵ = 1/2 with ϵ' = 0.4). The argument is easily repaired by choosing e large enough that c/e + ϵ' ≤ ϵ, but the stated choice should be corrected.
[Section 1, Theorem 1.7] Theorem 1.7 states 'Fix a finite surjective morphism π:Y→X' and then imposes the splitting hypothesis K_{(Y,D0)}^{⊗m} ≃ M^{⊗n}⊗N on that fixed cover. As stated this is vacuous unless such a cover exists; the theorem should either state that there exists a finite surjective morphism π satisfying (5.2)-(5.3), or clarify that the conclusion is conditional on the chosen π satisfying the splitting.
[Section 5, proof of Theorem 5.3] The derivation of (5.8) from (5.5), (5.6), and (5.7) is compressed. Spelling out the chain — h_{K(Y,D0)} = (n/m)h_M + (1/m)h_N ≥ (n/m)h_M - O(1), so (5.7) yields (n/m)(1-ϵ)h_M ≤ d_K + O(1), and multiplying by m gives exactly the coefficient n(1-ϵ) - m m_0(2r-2) in (5.8) — would remove any doubt that (5.8) is consistent with (5.3).

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The paper

The paper claims

The paper proves that Vojta's bounded-degree inequality with discriminant term, Levin's admissible-pairs inequality, and its truncated form are logically equivalent, and derives a conditional Bombieri-Lang statement for Deligne-Mumford…

The reading

The paper claims that three conjectural height inequalities in Diophantine geometry — Vojta's bounded-degree inequality with a discriminant term, Levin's admissible-pairs inequality, and its truncated form — are logically equivalent, so that proving any one of them proves all three. The difficult direction is the 'harder implication' that the admissible-pairs inequality implies the truncated form, proved by pulling back to a ramified cover where counting functions become tractable. As a conditional application, the paper shows that for nonsingular Deligne-Mumford stacks whose log canonical bundle is nearly a power of an ample bundle, Vojta's conjecture would imply a Bombieri-Lang statement: the integral points of bounded degree are not Zariski dense. A sympathetic reader should care because the equivalence reduces three separate conjectural statements to a single target, and the stack application extends the reach of Vojta's framework to orbifold-like spaces.

What carries the argument

The engine is the ramified $e$-fold cover $\pi: X' \to X$ of Lemma 3.6, which combines Vojta's Lemmas 3.3 and 3.4: it takes $e$-th roots of functions $f_i$ whose divisors are $A - D_i$, so that all components of $\pi^*D_1$ have multiplicity at least $e$. On this cover, the truncated counting function $n_S^{(1)}(D_1, x)$ dominates $(e-1)n_S^{(1)}(D_1', x')$, and the discriminant transform is governed by condition (3.2), so the two sides of the desired inequality can be matched. The second engine is Lemma 5.1, a Silverman-type bound $d_K(y) \le (2r-2)m_0 h_M(y) + O(1)$ that turns unbounded height into a discriminant bound, which combines with the splitting hypothesis $K_{(Y,D_0)}^{\otimes m} \simeq M^{\otimes n} \otimes N$ with $n > m m_0 (2r-2)$ to produce finiteness via the Northcott property.

Extended reading notes

Core claim

The central claim is stated as Theorem 1.4: Conjecture 1.1 (Vojta's bounded-degree inequality with discriminant term), Conjecture 1.2 (the admissible-pairs inequality for $(D_0,S)$-integral points), and Conjecture 1.3 (its truncated form) are pairwise logically equivalent. The genuinely new step is the harder implication, Conjecture 1.2 implying Conjecture 1.3, which is established by adapting Vojta's ramified-cover approach to the abc conjecture: one pulls back through a cover where all components of $\pi^*D_1$ have multiplicity at least $e$, uses the discriminant-transformation inequality of Lemma 3.7, and matches the truncated counting function on $X$ against the full counting function on $X'$. The paper further proves (Theorems 1.7 and 5.3) that, conditionally on Conjecture 1.1, certain general-type nonsingular Deligne-Mumford stacks with projective coarse moduli space have a non-Zariski-dense set of $(D_0,S)$-integral points of bounded degree, a form of the Bombieri-Lang conjecture for stacks.

Load-bearing premise

The load-bearing premise is condition (3.2): after enlarging the finite set of places $S$, the discriminant gap $d_{K,S}(x') - d_{K,S}(x)$ between a bounded-degree point and its lift to the ramified cover must be bounded by $(e-1)n_S^{(1)}(D', x')$, a condition the paper justifies in a single sentence by citing Hermite's discriminant theorem without supplying the argument. If that inequality cannot always be met, the equivalence theorem collapses at its hard direction.

Editorial extensions

If this is right

Conjectures 1.1, 1.2 and 1.3 stand or fall together: a proof of any one of the three yields the other two, in the spirit in which the abc Conjecture and Vojta's inequality were meant to be interlocked.
If Conjecture 1.1 holds, then for nonsingular Deligne-Mumford stacks with projective coarse moduli space whose log canonical bundle is big and splits as $K_{(X,D_0)}^{\otimes m} \simeq M^{\otimes n} \otimes N$ with $n > m m_0(2r-2)$, the set of $(D_0,S)$-integral points of degree at most $d$ is not Zariski dense.
Working on the finite cover of Theorem 1.5 avoids the stack-theoretic discriminant term and the absence of a Northcott property for big line bundles that arise when formulating height inequalities directly on the underlying stack.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Condition (3.2) is the point most worth probing: if the Hermite-discriminant justification in Remark 3.8 cannot be turned into an unconditional argument for the specific covers of Lemma 3.6, then Theorem 1.4 still gives the equivalence of Conjectures 1.1 and 1.2, but the truncated-form equivalence in Conjecture 1.3 would hang unresolved.
The splitting hypothesis $n > m m_0(2r-2)$ confines the Bombieri-Lang application to log pairs whose log canonical bundle is very nearly a power of an ample line bundle; the genuinely general-type pairs, where the log canonical bundle is big but not nearly ample, lie outside the scope, and extending the argument there is a natural next step.
The Silverman-type bound of Lemma 5.1 is explicitly non-optimal (Remark 5.2), so the constant $(2r-2)$ that forces the strong splitting hypothesis might be improvable; tightening it would directly widen the class of stacks covered by Theorem 1.7.

Original abstract

We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a \emph{harder implication} which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and V\'{a}rilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.

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We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a \emph{harder implication} which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and V\'{a}rilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.

background contribution method result interpretation main claim

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s0	We expand on the concept of \emph{admissible pairs} from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}.	contribution 0.79	0.79	0.05	asserted 0.91
s1	In this context, among other new results, we prove a \emph{harder implication} which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}).	result 0.69	0.95	0.09	asserted 1.00
s2	As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree.	result 0.99	0.95	0.96	asserted 1.00
s3	A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and V\'{a}rilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}.	method 0.77	0.04	0.02	asserted 1.00
s4	Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights.	method 0.62	0.04	0.02	asserted 1.00
s5	As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.	contribution 0.48	0.50	0.02	asserted 0.62

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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References

Works cited (26)

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D. Abramovich and J. Harris,Abelian varieties and curves inW d(C), Compositio Math.78 (1991), no. 2, 227–238

View on Pith 1991
[2]

D. Abramovich and A. V´ arilly-Alvarado,Level structures on abelian varieties and Vojta’s conjecture, Compositio Math.153(2017), no. 2, 373–394

View on Pith 2017
[3]

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,Geometry of algebraic curves. Vol. I, Springer-Verlag, 1985

View on Pith 1985
[4]

K. Ascher and A. Javanpeykar,Bounding heights uniformly in families of hyperbolic varieties, New York J. Math.23(2017), 1791–1808. 16 NATHAN GRIEVE

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E. Bombieri and W. Gubler,Heights in Diophantine geometry, Cambridge University Press, Cambridge, 2006

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O. Debarre and R. Fahlaoui,Abelian varieties inW r d (C)and points of bounded degree on algebraic curves, Compositio Math.88(1993), no. 3, 235–249

View on Pith 1993
[7]

J. S. Ellenberg, M. Satriano, and D. Zureick-Brown,Heights on stacks and a generalized Batyrev-Manin-Malle conjecture, Forum Math. Sigma11(2023), Paper No. e14, 54

View on Pith 2023
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G. Faltings,Diophantine approximation on abelian varieties, Ann. of Math. (2)133(1991), no. 3, 549–576

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G. Frey,Curves with infinitely many points of fixed degree, Israel J. Math.85(1994), no. 1-3, 79–83

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N. Grieve,Segre embeddings and the canonical image of a curve, Geom. Dedicata164(2013), no. 1, 1–7

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[11]

,On arithmetic inequalities for points of bounded degree, Res. Number Theory7 (2021), no. 1, Paper No. 1, 14

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,Effective calculation of local Weil functions via presentations of Cartier divisors, Houston J. Math.51(2025), no. 2, 231–245

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,On qualitative aspects of the quantitative subspace theorem, Rocky Mountain J. Math.55(2025), no. 1, 121–146

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[14]

N. Grieve and C. Noytaptim,On relative fields of definition for log pairs, Vojta’s height inequalities and asymptotic coordinate size dynamics, Taiwanese J. Math.29(2025), no. 4, 715–735

View on Pith 2025
[15]

J. Harris and J. Silverman,Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. (1991), no. 2, 347–356

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J. Koll´ ar and S. Mori,Birational geometry of algebraic varieties, Cambridge University Press, Cambridge, 1998

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A. Kresch and A. Vistoli,On coverings of Deligne-Mumford stacks and surjectivity of the Brauer map, Bull. London Math. Soc.36(2004), no. 2, 188–192

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G. Le,A Wirsing-type theorem for numerically equivalent divisors, Kodai Math. J.46(2023), no. 2, 228–241

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A. Levin,On the Schmidt subspace theorem for algebraic points, Duke Math. J.163(2014), no. 15, 2841–2885

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G. Smith and I. Vogt,Low degree points on curves, Int. Math. Res. Not. IMRN (2022), no. 1, 422–445

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P. Vojta,A more generalabcconjecture, Internat. Math. Res. Notices (1998), no. 21, 1103– 1116. Department of Mathematics, National Taiwan University, Astronomy and Mathemat- ics Building 5F, No. 1, Sec. 4, Roosevelt Rd., Taipei 10617, Taiwan (R.O.C.); School of Mathematics and Statistics, 4302 Herzberg Laboratories, Carleton University, 1125 Colonel By Dr...

View on Pith 1998

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. Abramovich and J. Harris,Abelian varieties and curves inW d(C), Compositio Math.78 (1991), no. 2, 227–238

View on Pith 1991

[2] [2]

D. Abramovich and A. V´ arilly-Alvarado,Level structures on abelian varieties and Vojta’s conjecture, Compositio Math.153(2017), no. 2, 373–394

View on Pith 2017

[3] [3]

E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris,Geometry of algebraic curves. Vol. I, Springer-Verlag, 1985

View on Pith 1985

[4] [4]

K. Ascher and A. Javanpeykar,Bounding heights uniformly in families of hyperbolic varieties, New York J. Math.23(2017), 1791–1808. 16 NATHAN GRIEVE

View on Pith 2017

[5] [5]

E. Bombieri and W. Gubler,Heights in Diophantine geometry, Cambridge University Press, Cambridge, 2006

View on Pith 2006

[6] [6]

O. Debarre and R. Fahlaoui,Abelian varieties inW r d (C)and points of bounded degree on algebraic curves, Compositio Math.88(1993), no. 3, 235–249

View on Pith 1993

[7] [7]

J. S. Ellenberg, M. Satriano, and D. Zureick-Brown,Heights on stacks and a generalized Batyrev-Manin-Malle conjecture, Forum Math. Sigma11(2023), Paper No. e14, 54

View on Pith 2023

[8] [8]

G. Faltings,Diophantine approximation on abelian varieties, Ann. of Math. (2)133(1991), no. 3, 549–576

View on Pith 1991

[9] [9]

G. Frey,Curves with infinitely many points of fixed degree, Israel J. Math.85(1994), no. 1-3, 79–83

View on Pith 1994

[10] [10]

N. Grieve,Segre embeddings and the canonical image of a curve, Geom. Dedicata164(2013), no. 1, 1–7

View on Pith 2013

[11] [11]

,On arithmetic inequalities for points of bounded degree, Res. Number Theory7 (2021), no. 1, Paper No. 1, 14

View on Pith 2021

[12] [12]

,Effective calculation of local Weil functions via presentations of Cartier divisors, Houston J. Math.51(2025), no. 2, 231–245

View on Pith 2025

[13] [13]

,On qualitative aspects of the quantitative subspace theorem, Rocky Mountain J. Math.55(2025), no. 1, 121–146

View on Pith 2025

[14] [14]

N. Grieve and C. Noytaptim,On relative fields of definition for log pairs, Vojta’s height inequalities and asymptotic coordinate size dynamics, Taiwanese J. Math.29(2025), no. 4, 715–735

View on Pith 2025

[15] [15]

J. Harris and J. Silverman,Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. (1991), no. 2, 347–356

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