Geometry of quintics in $\mathbb P^3$ and the Craighero-Gattazzo surface of general type

Kalyan Banerjee

arxiv: 1906.09617 · v1 · pith:JOBLK636new · submitted 2019-06-23 · 🧮 math.AG

Geometry of quintics in mathbb P³ and the Craighero-Gattazzo surface of general type

Kalyan Banerjee This is my paper

Pith reviewed 2026-05-25 17:43 UTC · model grok-4.3

classification 🧮 math.AG MSC 14J2914E05

keywords Craighero-Gattazzo surfacetri-canonical systembase point freesingular quinticelliptic singularitiesnon-rationalityinvolutionsurfaces of general type

0 comments

The pith

The tri-canonical system on the Craighero-Gattazzo surface is base point free and separates tangent vectors at certain points; the normalization of the quotient of a general curve from a singular quintic by an involution is non-rational.

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

This paper examines whether the tri-canonical linear system on the Craighero-Gattazzo surface of general type is base point free and identifies the points where it separates tangent vectors. It also investigates the birational geometry of a singular quintic hypersurface in projective 3-space that carries four elliptic singularities. The central question is whether the normalization of the quotient of a general curve from the product of the cubic and quadric linear systems, taken under a given involution, is a non-rational variety. A reader would care because these statements bear directly on the existence of morphisms from surfaces of general type and on the construction of non-rational threefolds from classical hypersurface data.

Core claim

The paper claims that the tri-canonical system on the Craighero-Gattazzo surface is base point free and separates tangent vectors at certain points. It further claims that the normalization of the quotient of a general curve, taken under the given involution inside the product linear system of cubics and quadrics on a singular quintic in P^3 with exactly four elliptic singularities, is non-rational.

What carries the argument

The tri-canonical linear system on the Craighero-Gattazzo surface together with the involution on curves lying in the product of the cubic and quadric systems on the singular quintic.

If this is right

The tri-canonical map from the Craighero-Gattazzo surface is a morphism that is an immersion at the points where it separates tangent vectors.
The quotient construction produces new examples of non-rational threefolds whose minimal models arise from classical quintic data.
The base-point-freeness statement implies that the surface admits a morphism to a projective space of dimension equal to the dimension of the tri-canonical space.
The non-rationality result gives a concrete obstruction to rationality for certain quotients of curves on quintics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same techniques might be applied to other linear systems on the same surface to produce further embeddings or rational maps.
If the involution can be varied in a family, the construction may yield a positive-dimensional moduli space of non-rational threefolds.
The result on tangent separation could be used to study the deformation theory of the surface via the geometry of its canonical image.

Load-bearing premise

The Craighero-Gattazzo surface and the singular quintic with exactly four elliptic singularities exist and carry the linear systems and involution described in the prior literature without extra fixed components.

What would settle it

An explicit computation exhibiting a base point of the tri-canonical system on the Craighero-Gattazzo surface or proving that the normalization of one such quotient is a rational variety.

read the original abstract

In this paper we study the question whether the tri-canonical system on the Craighero-Gattazzo surface is base point free and at which points does it separate tangent vectors. Also we study the non-rationality of the normalization of the quotient of a general curve (under a given involution) in the product linear system of cubics and quadrics on a singular quintic in $\PR^3$ with four elliptic singularities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers two narrow calculations on the Craighero-Gattazzo surface and a quintic quotient but both rest on unverified properties taken from earlier references.

read the letter

This paper checks whether the tri-canonical system on the Craighero-Gattazzo surface is base-point-free and separates tangents at some points, and it shows that the normalization of the quotient of a general curve from the |3H| + |2H| system on a singular quintic with four elliptic singularities is non-rational. Those are the two concrete outputs. It applies standard facts about these objects and the given involution to reach the stated conclusions. The calculations follow the usual pattern for linear systems and quotients in this corner of algebraic geometry. The soft spot is exactly the one in the stress-test note: both claims are conditional on the Craighero-Gattazzo surface having the expected invariants and on a general member of that linear system on the quintic having precisely four elliptic singularities with the stated involution acting without extra fixed loci. The paper does not re-derive or independently check those base objects, so any gap in the cited constructions would make the new statements inapplicable to the intended examples. No new construction or verification appears. This is narrow, example-driven work aimed at people already studying surfaces of general type or linear systems on quintics in P^3. A reader deep in those examples could find the base-point-freeness or non-rationality statements useful for their own bookkeeping, but the paper does not introduce methods or resolve wider questions. I would send it to peer review. The claims are precise enough that area referees can check the derivations against the cited literature.

Referee Report

2 major / 2 minor

Summary. The paper claims that the tri-canonical system on the Craighero-Gattazzo surface is base-point-free and separates tangent vectors at certain points, and that the normalization of the quotient of a general curve (under a given involution) in the product linear system of cubics and quadrics on a singular quintic in P^3 with four elliptic singularities is non-rational.

Significance. If the results hold, the work supplies concrete geometric data on the linear systems of a known surface of general type and on quotients arising from curves on singular quintics. These details could support classification efforts and rationality questions in algebraic geometry. The manuscript builds directly on prior constructions rather than re-deriving the key objects.

major comments (2)

[Introduction] Introduction and abstract: the base-point-freeness of |3K|, the tangent-separation statements, and the non-rationality of the quotient normalization are all conditional on the Craighero-Gattazzo surface and the singular quintic (with precisely four elliptic singularities) existing with the invariants and properties asserted in the cited prior literature; the manuscript performs no independent reconstruction or verification of these objects.
[Section on the involution and quotient] Section describing the involution: the claim that the normalization of the quotient is non-rational assumes the given involution on the general curve in |3H|+|2H| acts without additional fixed points or irreducible components; no explicit local equations or fixed-locus computation is supplied to confirm this.

minor comments (2)

Notation for projective 3-space alternates between P^3 and PR^3; adopt a single consistent symbol.
Add precise theorem or proposition citations (with page numbers) from the referenced papers when invoking the existence, invariants, or singularity type of the Craighero-Gattazzo surface and the quintic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and constructive feedback. We address the major comments point by point below, clarifying the scope of our contributions and indicating where revisions will be made.

read point-by-point responses

Referee: [Introduction] Introduction and abstract: the base-point-freeness of |3K|, the tangent-separation statements, and the non-rationality of the quotient normalization are all conditional on the Craighero-Gattazzo surface and the singular quintic (with precisely four elliptic singularities) existing with the invariants and properties asserted in the cited prior literature; the manuscript performs no independent reconstruction or verification of these objects.

Authors: We acknowledge that our results on |3K| and the quotient normalizations are conditional on the existence and properties of the Craighero-Gattazzo surface and the singular quintic as established in the cited prior literature. The manuscript does not attempt an independent reconstruction, as its purpose is to investigate new properties (base-point-freeness, tangent separation, and non-rationality of quotients) building directly on these known constructions. This is standard practice for work on specific examples in algebraic geometry. revision: no
Referee: [Section on the involution and quotient] Section describing the involution: the claim that the normalization of the quotient is non-rational assumes the given involution on the general curve in |3H|+|2H| acts without additional fixed points or irreducible components; no explicit local equations or fixed-locus computation is supplied to confirm this.

Authors: The referee is correct that the non-rationality claim relies on the involution having no unexpected fixed points or components on a general curve, and that the manuscript provides no explicit local equations for the fixed locus. While this is expected from the generality of the curve in the linear system, we agree that supplying a brief local computation would strengthen the argument. We will add such a computation (using local equations at potential fixed points) in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims conditional on external prior constructions

full rationale

The paper assumes existence and basic properties of the Craighero-Gattazzo surface and the singular quintic (with four elliptic singularities) from earlier references, then studies base-point-freeness of |3K| and non-rationality of a quotient normalization. No equations, fitted parameters, or self-referential derivations appear in the provided text. The central claims do not reduce to the paper's own inputs by construction, nor rely on load-bearing self-citations or ansatzes smuggled from the author's prior work. This is a standard conditional geometric study against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information is given on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5592 in / 1163 out tokens · 30575 ms · 2026-05-25T17:43:45.216589+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · 3 internal anchors

[1]

\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

K. Banerjee and V. Guletskii, Rational equivalence for line configurations on cubic hypersurfaces in P^5 . , arXiv:1405.6430v1 , 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014
[2]

P.Craighero, R.Gattazzo, Quintics surfaces in ^3 having a non-singular model with q=p_g=0 , P_2 0 ,Ren. Sem. Math. Uni. Padova, 91:187-198,1994

work page 1994
[3]

A simply connected numerical Godeaux surface with ample canonical class

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. , arXiv:alg-geom/9704022

work page internal anchor Pith review Pith/arXiv arXiv
[4]

Y.Miyaoka, Tricanonical map of numerical Godeaux surfaces , Inventiones Mathematicae, 99-111,1976

work page 1976
[5]

Annalen,304, 151-155,1996

O.K\" u chle, Ample line bundles on blown-up surfaces , Math. Annalen,304, 151-155,1996

work page 1996
[6]

C.Voisin, Symplectic invoultions of K 3 surfaces act trivially on CH_0 , Documenta Mathematicae 17, 851-860, 2012

work page 2012
[7]

K.Banerjee, Jaya NN Iyer, On the kernel of the push-forward homomorphism between Chow groups , arxiv::1504.07887v1 ,2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
[8]

, unpublished, 2015

K.Banerjee, Collino's theorem for Higher dimensional varieties and its' applications. , unpublished, 2015

work page 2015
[9]

Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

A. Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

work page 1986
[10]

C.Birkenhake and H.Lange, Complex abelian varieties , Springer-Verlag Berlin, Heidelberg, New York, 2003

work page 2003
[11]

Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

S. Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

work page 1986
[12]

S.Bloch, An example in the theory of algebraic cycles , Algebraic K-Theory, Lecture notes in Mathematics, Volume 551, 1-29

work page
[13]

Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

A. Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

work page 1975
[14]

Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2

W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer-Verlag, Berlin, 1984

work page 1984
[15]

Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No

R. Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977

work page 1977
[16]

Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

C. Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

work page 2003
[17]

K.Banerjee, One dimensional algebraic cycles on non-singular cubic fourfolds in ^5 , PhD Thesis, University of Liverpool, 2014

work page 2014

[1] [1]

\'Etale monodromy and rational equivalence for $1$-cycles on cubic hypersurfaces in $\mathbb P^5$

K. Banerjee and V. Guletskii, Rational equivalence for line configurations on cubic hypersurfaces in P^5 . , arXiv:1405.6430v1 , 2014

work page internal anchor Pith review Pith/arXiv arXiv 2014

[2] [2]

P.Craighero, R.Gattazzo, Quintics surfaces in ^3 having a non-singular model with q=p_g=0 , P_2 0 ,Ren. Sem. Math. Uni. Padova, 91:187-198,1994

work page 1994

[3] [3]

A simply connected numerical Godeaux surface with ample canonical class

I.Dolgachev, C.Werner, A simply connected numerical Godeaux surface with ample canonical class. , arXiv:alg-geom/9704022

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Y.Miyaoka, Tricanonical map of numerical Godeaux surfaces , Inventiones Mathematicae, 99-111,1976

work page 1976

[5] [5]

Annalen,304, 151-155,1996

O.K\" u chle, Ample line bundles on blown-up surfaces , Math. Annalen,304, 151-155,1996

work page 1996

[6] [6]

C.Voisin, Symplectic invoultions of K 3 surfaces act trivially on CH_0 , Documenta Mathematicae 17, 851-860, 2012

work page 2012

[7] [7]

K.Banerjee, Jaya NN Iyer, On the kernel of the push-forward homomorphism between Chow groups , arxiv::1504.07887v1 ,2015

work page internal anchor Pith review Pith/arXiv arXiv 2015

[8] [8]

, unpublished, 2015

K.Banerjee, Collino's theorem for Higher dimensional varieties and its' applications. , unpublished, 2015

work page 2015

[9] [9]

Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

A. Beauville, Sur l'anneau de Chow d'une vari\'et\'e ab\'elienne , Mathematische Annalen, 1986

work page 1986

[10] [10]

C.Birkenhake and H.Lange, Complex abelian varieties , Springer-Verlag Berlin, Heidelberg, New York, 2003

work page 2003

[11] [11]

Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

S. Bloch, Algebraic cycles and higher K-theory , Advances in Mathematics, 61, 267-304, 1986

work page 1986

[12] [12]

S.Bloch, An example in the theory of algebraic cycles , Algebraic K-Theory, Lecture notes in Mathematics, Volume 551, 1-29

work page

[13] [13]

Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

A. Collino, The rational equivalence ring of symmetric product of curves , Illinois journal of mathematics,19,no.4, 567-583, 1975

work page 1975

[14] [14]

Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2

W. Fulton, Intersection theory , Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 2. Springer-Verlag, Berlin, 1984

work page 1984

[15] [15]

Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No

R. Hartshorne, Algebraic geometry , Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977

work page 1977

[16] [16]

Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

C. Voisin, Complex algebraic geometry and Hodge theory II , Cambridge University Press, Cambridge, 2003

work page 2003

[17] [17]

K.Banerjee, One dimensional algebraic cycles on non-singular cubic fourfolds in ^5 , PhD Thesis, University of Liverpool, 2014

work page 2014