The modified diagonal cycle on the triple product of a pointed curve

Gross, Benedict H.; Schoen, Chad

doi:10.5802/aif.1469

Cited by 62 publications

(101 citation statements)

References 3 publications

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“…As already shown by Gross and Schoen ( [Gro95]), this model can be desingularized to a regular strict semistable scheme. Using as additional data an ordering on the set X (0) s of irreducible components of X s we can make this desingularization canonical, therefore we get a well-defined desingularization W of the scheme X d .…”

mentioning

confidence: 80%

“…In [Gro95], Gross and Schoen investigate products of regular strict semi-stable schemes in general. The same procedure is later described by Hartl [Har01] together with a desingularization for X × S Spec R n , where X is any regular strict semi-stable scheme.…”

Section: Desingularizationmentioning

confidence: 99%

“…We use a desingularization similar to [Gro95,Prop 6.11] and [Har01] to get a regular strict semi-stable model W (X, <, d) of (X η ) d . The result of this process depends on a sequence of components, which is chosen arbitrarily in the citations above.…”

Section: Desingularizationmentioning

confidence: 99%

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A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

Kolb

2016

Abh. Math. Semin. Univ. Hambg.

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Abstract. We analyse the subring of the Chow ring with support generated by the irreducible components of the special fibre of the Gross-Schoen desingularization of a d-fold self product of a semi-stable curve over the spectrum of a discrete valuation ring. For this purpose we develop a calculus which allows to determine intersection numbers in the special fibre explicitly. As input our simplicial calculus needs only combinatorial data of the special fibre. It yields a practical procedure for calculating even self intersections in the special fibre. The first ingredient of our simplicial calculus is a localization formula, which reduces the problem of calculating intersection numbers to a special situation. In order to illustrate how our simplicial calculus works, we calculate all intersection numbers between divisors with support in the special fibre in dimension three and four. The localization formula and the general idea were already presented for d = 2 in a paper of Zhang [Zha10, Ch. 3]. In our present work we achieve a generalisation to arbitrary d.

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confidence: 80%

Section: Desingularizationmentioning

confidence: 99%

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A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

Kolb

2016

Abh. Math. Semin. Univ. Hambg.

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“…In the case when the base is a point the classes Γ n (C) are the modified diagonal classes; see [7], and [18]. For example, Γ 1 (C) = [C], and modulo ψ we have…”

Section: Tautological Classesmentioning

confidence: 99%

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

Moonen

Polishchuk

2010

J. Inst. Math. Jussieu

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Abstract. Let C be a family of curves over a non-singular variety S. We study algebraic cycles on the relative symmetric powers C [n] and on the relative Jacobian J. We consider the Chow homology CH * (C [•] /S) := ⊕n CH * (C [n] /S) as a ring using the Pontryagin product. We prove that CH * (C [•] /S) is isomorphic to CH * (J/S)[t] u , the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over CH * (J/S). We give two such isomorphisms that over a general base are different. Further we give precise results on how CH * (J/S) sits embedded in CH * (C [•] /S) and we give an explicit geometric description of how the operators ∂ [m] t and ∂u act. This builds upon the study of certain geometrically defined operators Pi,j (a) that was undertaken by one of us in Part 1 of this work, [18].Our results give rise to a new grading on CH * (J/S). The associated descending filtration is stable under all operators [N ] * and [N ] * acts on gr m Fil as multiplication by N m . Hence, after − ⊗ Q this filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's.Finally we give a version of our main result for tautological classes, and we show how our methods give a simple geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis, as later refined by one of us in [14].

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“…Then ∆ e is a cohomologically trivial cycle on X 3 , studied in detail by B. Gross and C. Schoen [11]. Assume that k is either a number field or a function field of a curve, and assume that X has semistable reduction over k. Then Gross and Schoen construct in [11] a canonical R-valued height (∆ e , ∆ e ) associated to ∆ e , fitting in a general approach due to A. Beilinson [4] and S. Bloch [5].…”

Section: Introductionmentioning

confidence: 99%

Normal functions and the height of Gross-Schoen cycles

Jong¹

2014

Nagoya Math. J.

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Abstract. We prove a variant of a formula due to S. Zhang relating the Beilinson-Bloch height of the Gross-Schoen cycle on a pointed curve with the self-intersection of its relative dualizing sheaf. In our approach the height of the Gross-Schoen cycle occurs as the degree of a suitable Bloch line bundle. We show that the Chern form of this line bundle is non-negative, and we calculate its class in the Picard group of the moduli space of pointed stable curves of compact type. The basic tools are normal functions and biextensions associated to the cohomology of the universal jacobian.

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The modified diagonal cycle on the triple product of a pointed curve

Cited by 62 publications

References 3 publications

A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

Normal functions and the height of Gross-Schoen cycles

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