Some remarks on modified diagonals

Moonen, B.J.; Yin, Qizheng

doi:10.1142/s0219199715500091

Cited by 4 publications

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References 15 publications

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“…Remark 4.11. Proposition 4.9 is also proved in [11], where it is used to deduce the vanishing Γ g+2 (X, a) = 0 of Remark 1.4, for any point a ∈ X, from the main result of Colombo and van Geemen [5].…”

Section: Case Of Curvesmentioning

confidence: 99%

Some new results on modified diagonals

Voisin

2015

Geom. Topol.

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O'Grady studied recently $m$-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles $\Gamma^m(X,a)$ depend on a choice of reference point $a\in X$ (or more generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle $\Gamma^m(X,a)$ vanishes for large $m$. We also prove the following conjecture of O'Grady: if $X$ is a double cover of $Y$ and $\Gamma^m(Y,a)$ vanishes (where $a$ belongs to the branch locus), then $\Gamma^{2m-1}(X,a)$ vanishes, and we provide a generalization to higher degree finite covers. We finally prove the vanishing $\Gamma^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$ a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him for $m=2,3$.Comment: Final version, to appear in Geometry and Topolog

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Section: Case Of Curvesmentioning

confidence: 99%

Some new results on modified diagonals

Voisin

2015

Geom. Topol.

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“…• In [Voi15], Voisin showed that if X is a smooth projective connected variety of dimension n swept out by irreducible curves of genus g supporting a zero-cycle rationally equivalent to pt ∈ X, then we have ∆ ′ m = 0 for m ≥ (n + 1)(g + 1). • In [MY16], Moonen and Yin showed that -∆ ′ n = 0 on a g-dimensional abelian variety precisely when n ≥ 2g + 1, and -∆ ′ n = 0 on a curve of genus g whenever n ≥ g + 2 (which is sharp for a generic pointed curve, see [Qiz14]). We will show that the minimal k ≤ m such that ∆ ′ k = 0 determines the Z-linear relations between the n m classes ∆ A (A) ∈ A • (X n ) Q for A ⊂ {1, .…”

Section: Introductionmentioning

confidence: 99%

Modified diagonals and linear relations between small diagonals

Spink¹

2020

Proc. Amer. Math. Soc.

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We prove that the vanishings of the modified diagonal cycles of Gross and Schoen govern the Z-linear relations between small m-diagonals pt {1,...,n}\A × ∆ A in the rational Chow ring of X n for A ranging over melement subsets of {1, . . . , n}. Our results generalize to arbitrary symmetric classes in place of the diagonal in X m , and with different types of inclusionsThe combinatorial heart of this paper, which may be of independent interest, is showing the Z-linear relations between elementary symmetric polynomials e k (xa 1 , . . . , xa m ) ∈ Z[x 1 , . . . , xn] are generated by the Sn-translates of a certain alternating sum over the facets of a hyperoctahedron.

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Tautological classes on the moduli space of hyperelliptic curves with rational tails

Tavakol

2018

Journal of Pure and Applied Algebra

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We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. The connection with recent conjectures by Pixton is also discussed. CONTENTS TAUTOLOGICAL CLASSES ON THE MODULI SPACE H rt g,n 3 Everything mentioned above concerns tautological classes in Chow. The Gorenstein property of R * (H rt g,n ) implies the same results in cohomology. This shows that there is no difference between Chow and cohomology as long as we restrict to tautological classes. Using a result of Petersen and Tommasi, which was our motivation for this project, we prove the following:Corollary 0.2. The cycle class map induces an isomorphism between the tautological ring of the moduli space H rt g,n in Chow and monodromy invariant classes in cohomology.At the end we discuss the connection between the relations on the space H rt g,n and Pixton's relations on M g,n . Conventions 0.3. We consider algebraic cycles modulo rational equivalence. All Chow groups are taken with Q-coefficients.

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Some remarks on modified diagonals

Abstract: We prove a number of basic vanishing results for modified diagonal classes. We also obtain some sharp results for modified diagonals of curves and abelian varieties, and we prove a conjecture of O'Grady about modified diagonals on double covers.

Cited by 4 publications

References 15 publications

Some new results on modified diagonals

Some new results on modified diagonals

Modified diagonals and linear relations between small diagonals

Tautological classes on the moduli space of hyperelliptic curves with rational tails

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