Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

Douglas Ulmer; Giancarlo Urz\'ua

REVIEW 1 major objections 5 minor 1 cited by

Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that, over a field of characteristic zero, the zero section of a Jacobian elliptic surface is tangent to nP for only finitely many base points; in very general families no tangencies occur at all.

desk verdict Genuinely new Betti-foliation results and a striking geography construction, but the proof of the central finiteness theorem has a load-bearing gap in the additive reduction cases. read the letter →

arxiv 1908.02208 v4 pith:K5BODT3Q submitted 2019-08-06 math.AG math.NT

Douglas Ulmer , Giancarlo Urzúa This is my paper · ORCID

classification math.AGmath.NT MSC 14J2711B3914J29

keywords ellipticsurfacessectionstransversalitydivisibilitysequencesBetticoordinatesgeographyofalgebraiclog-terminalsingularitiesunlikelyintersections

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper studies a Jacobian elliptic surface π:E→C with a zero section O and another section P of infinite order. It proves that the set of points of C where O is tangent to some multiple nP is finite, and equivalently that the elliptic divisibility divisors Dn=O∗(nP) are reduced for every n not divisible by one of a finite set of integers. For a very general choice of coefficients in a sufficiently positive linear system, it proves the stronger statement that O meets nP transversally in exactly d(n2−1) points for every n prime to the characteristic. This transversality drives an explicit geography result: there exist normal surfaces of any fixed geometric genus with one mild singular point, ample canonical class, and arbitrarily large K2.

What carries the argument

The load-bearing object is the local Betti foliation: over a disk in the good-reduction locus, the elliptic surface is (Δ×C)/(Zτ+Z), and the leaves Lr,s are the images of z↦(z,rτ(z)+s). In the real coordinates (r,s) on the universal cover, every torsion section becomes a constant rational leaf, and the section P becomes a real-analytic map φ:Δ→R2 whose derivative vanishes exactly at tangencies with some leaf. The Lojasiewicz gradient inequality then rules out accumulating zeros of the derivative away from the point where φ equals its base value, giving discreteness. Near bad fibers, Kodaira's models convert the same idea into an inequality comparing zf′(z)/f(z) with log|f(z)|/log|z|, proving no tangencies can accumulate at a multiplicative fiber. The other main mechanism is the moduli space of triples (E,ω,P), represented by Spec k[a2,a3,a4], where the condition P∈E[n] is a hypersurface; the very-general transversality theorem follows by showing the incidence correspondence over V=H0(L2⊕L3⊕L4) is smooth and étale over a nonempty open parameter set.

What would settle it

Take a concrete elliptic surface with one additive potentially good fiber and a non-torsion section P, pull back by the asserted ramified cover of degree 2, 3, 4, or 6, and compute the tangency points of P with the torsion multisections E[n] near the ramification point. If those tangencies accumulate at the fiber, the omitted case in Section 3 collapses the discreteness argument; if the pulled-back equation lands in the already-proved I0, I1, Ib, or I*b cases with no accumulation, the load-bearing step is confirmed.

Watch

Extended reading notes

Core claim

The paper's central claim is that for a relatively minimal Jacobian elliptic surface over the complex numbers, with zero section O and a non-torsion section P, the set T=⋃n≠0{t∈C:nP is tangent to O at t} is finite. The proof works with the n-torsion multisection E[n]: P is tangent to E[n] exactly when nP is tangent to O, so proving P is transverse to all E[n] away from finitely many points proves the theorem. The authors show that the larger set of tangencies of P with all local Betti leaves Lr,s is discrete, hence finite, using Kodaira's local models and the Lojasiewicz gradient inequality; the torsion tangencies are contained in this larger set. The same finiteness is equivalent to a square-freeness statement for function-field elliptic divisibility sequences: a finite set M exists such that Dn is reduced if and only if n is not divisible by any element of M. For very general data in H0(C,L2⊕L3⊕L4), the paper proves there are no tangencies at all, and applies this to construct, for every g≥0 and every N, a normal projective surface with geometric genus g, one log-terminal singular point, K ample and Q-Cartier, and K2>N.

Load-bearing premise

The proof relies on an unverified local step: near the six exceptional types of singular fiber with additive potentially good reduction, the authors assert that a ramified base change of degree 2, 3, 4, or 6 reduces the problem to cases already proved, and they leave the details as an exercise for the reader.

Editorial extensions

If this is right

For every non-torsion section in characteristic zero, the function-field elliptic divisibility sequence Dn has bounded 'non-square-free part': the degree of gcd(fn,dfn/dt) is bounded, not merely subquadratic.
For a very general parameter in H0(C,L2⊕L3⊕L4), every multiple nP meets O transversally in exactly d(n2−1) points, so the full sequence of sections is uniformly transverse.
There exist explicit pairs (E,P) over number fields and global function fields, with singular fibers of type I0*, in which nP meets O transversally in the exact counts (n2−1)/2 for odd n and (n2−4)/2 for even n.
Given any integers g≥0 and N, there is a normal projective surface with pg=g, one log-terminal singular point, ample K, and K2>N, so K2 has no upper bound in terms of pg among mildly singular surfaces of general type.

Reading between the lines

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

If the omitted base-change verification for the six additive potentially-good fiber types is completed, the same Betti-coordinate method should yield effective upper bounds on the number of tangencies, since the Lojasiewicz inequality produces quantitative separation near each fiber.
The very-general transversality statement suggests a concrete algebraic test in positive characteristic: for p>3 one expects the same transversality to hold for all n prime to p, and the moduli-space argument in the paper could likely be adapted without passing through complex analysis.
The geography construction implicitly answers a question about accumulation of volumes in moduli of surfaces: fixing the geometric genus does not bound the canonical self-intersection even for surfaces with only one log-terminal singularity, so any lower-bound results of that type cannot extend to upper bounds.
The Betti-leaf viewpoint may also yield a constructive algorithm for the finite set M: instead of checking each n separately, one could track the leaves met by P and compute the finitely many rational leaf parameters where the derivative vanishes.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

Read this one if you care about unlikely intersections on elliptic surfaces or geography of surfaces. The headline is that the paper has a real main theorem, a smart method, and one honest gap in the proof that should be fixed before the result is taken as established.

What is new: the tangency finiteness theorem (equivalent to a finiteness statement for non-reduced elliptic divisibility divisors) was independently obtained by Corvaja-Demeio-Masser-Zannier; the authors say so. What they add is the Betti-foliation proof, which is a genuinely different and more geometric route. The new material is the very general no-tangency results (Theorems 1.7/1.8) and the geography application (Theorem 1.9): log-terminal surfaces of fixed geometric genus with K ample and K^2 unbounded. That is a striking answer to a natural boundedness question, and the construction is explicit and checks out. The explicit examples over number fields (Section 7) are also valuable and self-contained. Credit where due: the paper is honest about the overlap and mostly writes down real arguments rather than deferring to references.

The soft spot is in the proof of the central Theorem 2.5. The cases I0, I1, Ib, I*b are written in detail, but the additive potentially good types II, II*, III, III*, IV, IV* are disposed of with 'we leave the details as an exercise for the reader.' That line is load-bearing: Claim 3.1 needs discreteness of tangencies near every fiber, including those. A ramified base change of degree 2, 3, 4, or 6 ought to reduce to the good case, but you still have to check that the Betti leaf tangency condition transforms correctly and that accumulation in the punctured disk is really ruled out. The paper does not do that check. I don't see an obvious reason it fails—the reduction is plausible and the authors know this material—but as written the main theorem has a hole. This is not a manufactured nitpick; it is an explicit gap flagged in the text.

The geography section engages with the theorem as a black box, and the theorem is plausible; if the gap is filled, the paper is strong. The citation pattern is fine. No circularity: the proof uses standard results, and the overlap with CDMZ is properly credited.

Who this is for: anyone working on elliptic surfaces, divisibility sequences, or surface geography. It deserves a serious referee; the right call is to send it out with a request that the additive case be written up. I would not accept it as is.

Best.

Referee Report

1 major / 5 minor

Summary. The paper studies a relatively minimal Jacobian elliptic surface E over a curve C in characteristic zero, equipped with a zero section O and a non-torsion section P. The central result (Theorem 1.1, reformulated as Theorem 2.5) asserts that the set of base points t at which O is tangent to nP for some nonzero integer n is finite, equivalently that the elliptic divisibility divisors D_n = O^*(nP) are reduced for all n outside a finite exceptional set. Section 3 proves this via local analysis of the Betti foliation: for good reduction and the multiplicative types I1, Ib, and I*_b, the proof derives explicit tangency equations and uses growth estimates to rule out accumulation of tangencies at the corresponding fibers. For the additive potentially good types II, II*, III, III*, IV, and IV*, the proof asserts a reduction to the good-reduction case by a ramified base change and leaves the details as an exercise. The paper then proves genericity and explicit-construction results (Theorems 1.7 and 1.8) in which no tangencies occur, and applies them to construct normal surfaces with fixed geometric genus, one log-terminal singularity, ample Q-Cartier canonical class, and arbitrarily large K^2 (Theorem 1.9).

Significance. If the main theorem is fully proved, this is a substantial contribution: it gives a sharp function-field analogue of square-freeness of elliptic divisibility sequences, replacing a conjectural o(n^2) bound by an actual finiteness statement; it establishes a non-obvious global transversality theorem through the Betti foliation; and it supplies new geography examples with unbounded K^2 for fixed geometric genus. The generic transversality theorem for very general data and the explicit height-2 examples over small fields are valuable in their own right. The paper is clearly written and uses standard tools (Kodaira models, Deligne-Rapoport, canonical height pairings, Artin contraction) in an appropriate way. However, the proof of the central finiteness theorem has a load-bearing gap in the additive reduction cases, so the manuscript is not yet suitable for publication as is.

major comments (1)

[Section 3, final case ('The cases of II, II*, III, III*, IV, and IV* reduction')] The proof of Theorem 2.5 is incomplete because the additive potentially good reduction types are dismissed with 'We leave the details as an exercise for the reader.' This is load-bearing: Claim 3.1 must exclude tangencies accumulating at every point of C, including at bad fibers. For the cases that are written out, the argument derives a concrete tangency equation (for example equation (3.3) for type I1) and uses the growth estimates in (3.5). For the additive types no such equation is derived, and the asserted reduction to good reduction after a ramified base change of order 2, 3, 4, or 6 requires several non-formal checks: that the pulled-back section stays outside the Betti foliation, that the tangency condition transforms correctly under the ramified cyclic cover, and that the absence of accumulation points on the cover implies the absence of accumulation in the original punctured disk. Since none of these checks appears in the manuscript, the discreteness argument does not currently cover points of types II, II*, III, III*, IV, or IV*, and Theorem 2.5, and therefore Theorems 1.1 and 1.4, are unproved as written.

minor comments (5)

[Section 3, first paragraph] The word 'compex manifolds' should be 'complex manifolds'.
[Remark 1.5] 'Thereom 1.4' should be 'Theorem 1.4'.
[Section 8, paragraph after blowing up] 'let Cj be the strict transform the section jP' should read 'let Cj be the strict transform of the section jP'.
[Proof of Theorem 1.8] In the sentence 'Composing φ with a linear fractional transformation', φ should be f, since f is the morphism C→P^1 just chosen.
[Statement of Theorem 1.8] The phrase 'there exist infinitely many pairs (E,P ) where E is a Jacobian elliptic surface E→C equipped with a section P such that' is grammatically awkward and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main finiteness theorem and the generic transversality results are derived from independent local analytic, moduli-theoretic, and height-theoretic inputs; the only flagged gap is an omitted reduction for additive fibers, which is a completeness concern rather than a circular step.

full rationale

The central claim, Theorem 2.5, is proved by complex-analytic local models: the I0 case uses the Lojasiewicz gradient inequality, the I1 case uses Kodaira's local normal form and elementary growth estimates, and the Ib and I*b cases are explicitly reduced to those arguments. None of these inputs contains the conclusion, and no parameter is fitted to the tangency set. The reformulation of Theorem 1.1 as Theorem 1.4 is a translation of definitions: D_n = O^*(nP), and non-reducedness of D_n is exactly tangency of nP with O; this equivalence is not a derivation of content from the conclusion. The generic statement Theorem 1.7 is proved through the incidence correspondence D_n = F^{-1}(M_k[n]) and a transversality/étaleness argument in the moduli space, with non-emptiness verified by explicit examples over P1; this is independent of the theorem being proved. The only substantive self-citation is [Ulm11, Prop. I.7.3] in Lemma 6.5, used to rule out p-torsion sections in positive characteristic; that is a cited prior theorem with its own stated hypotheses, independent of the present conclusion, so under the review rules it counts as independent evidence rather than circularity. The paper does contain an explicit deferred case in the proof of Theorem 2.5: for additive potentially good fibers II, II*, III, III*, IV, IV*, it says 'We leave the details as an exercise for the reader.' This is a genuine proof gap if the ramified-base-change reduction is not verified, and it is load-bearing for the discreteness argument near those fibers. However, omitting a verification is not the same as assuming the theorem: the omitted step does not make the derivation circular, because it does not presuppose finiteness of Ttor or invoke the result being proved. Accordingly, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper does not introduce free parameters or invented entities. The proofs rest on standard results from the theory of elliptic surfaces, complex analysis, and moduli theory: Kodaira's local normal forms, the Lojasiewicz gradient inequality, Deligne-Rapoport's description of torsion divisors on the moduli stack, the height pairing machinery of Cox-Zucker and Shioda, Artin's contractibility theorem, and the Nakai-Moishezon criterion. The authors also use a prior theorem of Ulmer on absence of p-torsion in positive characteristic. None of these are ad hoc assumptions tailored to the target results.

assumptions (6)

standard math Kodaira's local normal forms and classification of elliptic surface fibers
Used throughout Section 3 to describe the surface near good and bad fibers and to define local Betti leaves.
standard math Lojasiewicz gradient inequality for real-analytic functions
Invoked in the I0 case of the proof of Theorem 2.5 to show the derivative of the Betti-coordinate map does not vanish away from the tangency point.
standard math Deligne-Rapoport representability of the moduli stack of elliptic curves with level structures and properties of torsion divisors
Used in Section 4 and in Proposition 6.6 to define the moduli space M and the codimension-one torsion locus M_k[n].
standard math Height pairing for sections of elliptic surfaces (Cox-Zucker, Shioda)
Provides the asymptotic formula (nP).O = ht(P)n^2/2 + O(1) in Lemma 2.8.
standard math Artin's contractibility theorem for curves on algebraic surfaces
Applied in Section 8 to contract the union C0 union Cn and form the singular surface X in Theorem 1.9.
standard math Ulmer's theorem on absence of p-torsion for non-isotrivial elliptic surfaces with j not a p-th power
Used in Lemma 6.5 to rule out p-torsion in positive characteristic p > 3.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces." pith.science (2026). https://pith.science/paper/K5BODT3Q

@misc{pith2026190802208,
  author       = {Pith},
  title        = {Pith review of: Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5BODT3Q}},
  note         = {Machine review of arXiv:1908.02208}
}

abstract

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with $K$ ample and $K^2$ unbounded.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the torsion values for sections of an elliptic scheme
math.AG 2019-09 conditional novelty 7.0 of 10

The canonical height of a section of an elliptic scheme over a curve equals the integral of the Betti form over the base, and this measure coincides with the dynamical equidistribution measure.

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · cited by 1 Pith paper

[2]

torsion multisections

P/r.sc/e.sc/l.sc/i.sc/m.sc/i.sc/n.sc/a.sc/r.sc/i.sc/e.sc/s.sc /o.sc/n.sc /t.sc/o.sc/r.sc/s.sc/i.sc/o.sc/n.sc /a.sc/n.sc/d.sc /i.sc/n.sc/t.sc/e.sc/r.sc/s.sc/e.sc/c.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section we gather various foundational results on torsion, inte rsections, heights, and elliptic divisibility sequences. Some of this material also appears in...

work page
[3]

Betti coordinates

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 2.5 We ﬁrst note that Theorem 2.5 is a statement about intersections on an elliptic surface over the complex numbers. To prove it, we may replaceE andC with the corresponding compex manifolds and make use of the classical topology, i.e., the topology induced by the metric topology on C. For ...

work page
[4]

the projective family of plane cubics deﬁned by y2 +a3y =x3 +a2x2 +a4x

I/n.sc/t.sc/e.sc/r.sc/l.sc/u.sc/d.sc/e.sc /o.sc/n.sc /m.sc/o.sc/d.sc/u.sc/l.sc/i.sc /o.sc/f.sc /e.sc/l.sc/l.sc/i.sc/p.sc/t.sc/i.sc/c.sc /c.sc/u.sc/r.sc/v.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /a.sc /d.sc/i.sc/f.sc/f.sc/e.sc/r.sc/e.sc/n.sc/t.sc/i.sc/a.sc/l.sc /a.sc/n.sc/d.sc /a.sc /p.sc/o.sc/i.sc/n.sc/t.sc In this section, we discuss certain moduli spaces of e...

work page
[5]

Letting γ be a square root of −c6/(4c4), the map to Gm is (x′,y ′)↦→y′−γx′ y′ +γx′ and we ﬁnd that P maps to a3c4−γ(16a2a4− 36a2 3) a3c4 +γ(16a2a4− 36a2

work page
[6]

(4.1) which (not surprisingly) is an algebraic expression in the original a2,a 3,a 4

work page
[7]

smallest

F/r.sc/o.sc/m.scE/K /t.sc/o.scE→C We remind the reader how to go from an elliptic curve over a function ﬁeld to an elliptic surface. Although this is not strictly necessary for our main purposes, it sug gests a fruitful point of view on ﬁnite-dimensional families of elliptic surfaces parameterized by certain Riemann-Roch spaces. 5.1. General construction....

work page
[8]

for a very general a

V/e.sc/r.sc/y.sc /g.sc/e.sc/n.sc/e.sc/r.sc/a.sc/l.sc /e.sc/l.sc/l.sc/i.sc/p.sc/t.sc/i.sc/c.sc /s.sc/u.sc/r.sc/f.sc/a.sc/c.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /t.sc/w.sc/o.sc /s.sc/e.sc/c.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section,k is a ﬁeld of characteristic zero orp> 3 andC is a smooth, projective, absolutely irreducible curve over k. Let L be a line bun...

work page
[9]

correction term

E/x.sc/p.sc/l.sc/i.sc/c.sc/i.sc/t.sc /e.sc/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /e.sc/v.sc/e.sc/n.sc /h.sc/e.sc/i.sc/g.sc/h.sc/t.sc /o.sc/v.sc/e.sc/r.sc /s.sc/m.sc/a.sc/l.sc/l.sc /f.sc/i.sc/e.sc/l.sc/d.sc/s.sc In this section, we show by explicit construction that there are pairs(E,P ) withP transverse to torsion multisections over ﬁeld...

work page

Show all 10 references

[10]

Let k = C,C = P1, and L = OP1(d) where d = g + 1, which by assumption satisﬁes d≥ 1

A/p.sc/p.sc/l.sc/i.sc/c.sc/a.sc/t.sc/i.sc/o.sc/n.sc /t.sc/o.sc /g.sc/e.sc/o.sc/g.sc/r.sc/a.sc/p.sc/h.sc/y.sc /o.sc/f.sc /s.sc/u.sc/r.sc/f.sc/a.sc/c.sc/e.sc/s.sc In this section, we will prove Theorem 1.9. Let k = C,C = P1, and L = OP1(d) where d = g + 1, which by assumption sa...

2019
[2019]

[CMZ18] P

Preprint, arxiv:1909.01253. [CMZ18] P. Corvaja, D. Masser, and U. Zannier, Torsion hypersurfaces on abelian schemes and Betti coordin ates, Math. Ann. 371 (2018), 1013–1045. [CZ79] D. A. Cox and S. Zucker, Intersection numbers of sections of elliptic surfaces , Invent. Math. 5...

2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

torsion multisections

P/r.sc/e.sc/l.sc/i.sc/m.sc/i.sc/n.sc/a.sc/r.sc/i.sc/e.sc/s.sc /o.sc/n.sc /t.sc/o.sc/r.sc/s.sc/i.sc/o.sc/n.sc /a.sc/n.sc/d.sc /i.sc/n.sc/t.sc/e.sc/r.sc/s.sc/e.sc/c.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section we gather various foundational results on torsion, inte rsections, heights, and elliptic divisibility sequences. Some of this material also appears in...

work page

[2] [3]

Betti coordinates

P/r.sc/o.sc/o.sc/f.sc /o.sc/f.sc T/h.sc/e.sc/o.sc/r.sc/e.sc/m.sc 2.5 We ﬁrst note that Theorem 2.5 is a statement about intersections on an elliptic surface over the complex numbers. To prove it, we may replaceE andC with the corresponding compex manifolds and make use of the classical topology, i.e., the topology induced by the metric topology on C. For ...

work page

[3] [4]

the projective family of plane cubics deﬁned by y2 +a3y =x3 +a2x2 +a4x

I/n.sc/t.sc/e.sc/r.sc/l.sc/u.sc/d.sc/e.sc /o.sc/n.sc /m.sc/o.sc/d.sc/u.sc/l.sc/i.sc /o.sc/f.sc /e.sc/l.sc/l.sc/i.sc/p.sc/t.sc/i.sc/c.sc /c.sc/u.sc/r.sc/v.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /a.sc /d.sc/i.sc/f.sc/f.sc/e.sc/r.sc/e.sc/n.sc/t.sc/i.sc/a.sc/l.sc /a.sc/n.sc/d.sc /a.sc /p.sc/o.sc/i.sc/n.sc/t.sc In this section, we discuss certain moduli spaces of e...

work page

[4] [5]

Letting γ be a square root of −c6/(4c4), the map to Gm is (x′,y ′)↦→y′−γx′ y′ +γx′ and we ﬁnd that P maps to a3c4−γ(16a2a4− 36a2 3) a3c4 +γ(16a2a4− 36a2

work page

[5] [6]

(4.1) which (not surprisingly) is an algebraic expression in the original a2,a 3,a 4

work page

[6] [7]

smallest

F/r.sc/o.sc/m.scE/K /t.sc/o.scE→C We remind the reader how to go from an elliptic curve over a function ﬁeld to an elliptic surface. Although this is not strictly necessary for our main purposes, it sug gests a fruitful point of view on ﬁnite-dimensional families of elliptic surfaces parameterized by certain Riemann-Roch spaces. 5.1. General construction....

work page

[7] [8]

for a very general a

V/e.sc/r.sc/y.sc /g.sc/e.sc/n.sc/e.sc/r.sc/a.sc/l.sc /e.sc/l.sc/l.sc/i.sc/p.sc/t.sc/i.sc/c.sc /s.sc/u.sc/r.sc/f.sc/a.sc/c.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /t.sc/w.sc/o.sc /s.sc/e.sc/c.sc/t.sc/i.sc/o.sc/n.sc/s.sc In this section,k is a ﬁeld of characteristic zero orp> 3 andC is a smooth, projective, absolutely irreducible curve over k. Let L be a line bun...

work page

[8] [9]

correction term

E/x.sc/p.sc/l.sc/i.sc/c.sc/i.sc/t.sc /e.sc/x.sc/a.sc/m.sc/p.sc/l.sc/e.sc/s.sc /w.sc/i.sc/t.sc/h.sc /e.sc/v.sc/e.sc/n.sc /h.sc/e.sc/i.sc/g.sc/h.sc/t.sc /o.sc/v.sc/e.sc/r.sc /s.sc/m.sc/a.sc/l.sc/l.sc /f.sc/i.sc/e.sc/l.sc/d.sc/s.sc In this section, we show by explicit construction that there are pairs(E,P ) withP transverse to torsion multisections over ﬁeld...

work page