[论文解读] Covers of curves, Ceresa cycles, and Unlikely intersections

研究动机与目标

• Motivate and study when the Ceresa cycle of covers of a curve is torsion or nontorsion in CH_1(J); connect this to broader questions about torsion loci in moduli.
• Develop and use the concept of a relative canonical shadow to translate Ceresa-torsion questions into Jacobian torsion questions.
• Employ unlikely intersection theory (Manin–Mumford–Raynaud, Masser–Zannier, and relative versions) to analyze the torsion locus in families of covers.
• Provide explicit families of genus 6 curves where the locus of torsion Ceresa cycles is Zariski closed with positive codimension.
• Relate findings to the structure of the Ceresa-torsion locus in moduli and to conjectures of Zilber–Pink type for variations of mixed Hodge structures.]
• method':['Reduce Ceresa-torsion questions to torsion questions for a related point on the curve’s Jacobian via the relative canonical shadow (Definition 1.1).','Use the intersection theory of cycles and pushforward relations to obtain a computable shadow in CH_0(C) and CH_1(J).','Apply the Manin–Mumford–Raynaud theorem and its relative version to constrain torsion translates within subvarieties of abelian schemes.','Leverage the Masser–Zannier unlikely intersections theorem in one- and two-parameter families to bound simultaneous torsion occurrences.','Utilize group actions (e.g., symmetric groups, dihedral groups) to produce elliptic fibrations and derive independent sections whose torsion behavior controls the Ceresa cycle.','Provide explicit 1- and 2-parameter genus 6 families to illustrate where the torsion locus is small and open dense where applicable.']
• research_questions':['Does the Ceresa cycle of a very general ramified cover of a genus g ≥ 2 curve remain nontorsion in CH_1(J)?','Can one relate Ceresa-torsion in covers to torsion properties of a naturally associated point on the curve’s Jacobian via the relative canonical shadow?','In families of covers, is the locus where the Ceresa cycle is torsion Zariski closed of positive codimension?','Can explicit low-dimensional families of curves be exhibited where the torsion locus is non-dense or has positive codimension?','How does the structure of the Hurwitz spaces and monodromy representations influence the torsion behavior of Ceresa cycles?]
• key_findings':['For very general ramified covers of a genus g ≥ 2 curve C, the Ceresa cycle of the covering curve is nontorsion in CH_1(J).','There exist infinitely many families of ramified covers with a very general member having a nontorsion Ceresa cycle.','Explicit 1- and 2-parameter genus 6 families show the locus of curves with torsion Ceresa cycle is Zariski closed and has positive codimension.','The relative canonical shadow provides a concrete obstruction to Ceresa triviality, translating a Ceresa-torsion question into a torsion problem for a Jacobian point.','Using relative Manin–Mumford and Masser–Zannier results, the authors establish that the torsion locus in these families is contained in a Zariski-closed subset of positive codimension.','The results contribute new examples where the Ceresa cycle is generically nontorsion and where the torsion locus is provably not dense in certain Hurwitz families.']
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