ON THE CHOW RING OF A K3 SURFACE

By A. Beauville. 2001

194 citations recorded by the source corpus.

We show that the Chow group of 0-cycles on a K3 surface contains a class of degree 1 with remarkable properties: any product of divisors is proportional to this class, and so is the second Chern class c2.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/582c1634989ae9104eec9dcc643dbd9c207c63a2

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Citations in the corpus

The Fourier Transform for Certain HyperKähler Fourfolds (2013, methodology): “This is due to the fact proved by Beauville–Voisin [11] that the intersection of any two divisors on X is a multiple of oX and was explained in the introduction.”
The generalized Franchetta conjecture for some hyper-Kähler varieties, II (2020, background): “ive varieties, there are examples of varieties that do not admit such a multiplicative decomposition; it was howeverconjectured [SV16a, Conjecture 4], following theseminal work of Beauville and Voisin[BV04], thatall hyper-Ka¨hlervarietiesadmit suchadecomposition. We referto[FLV19]foralistofhyper-Ka¨”
Lectures on K3 Surfaces (2016, background): “We only present an outline of the main arguments and refer to [52] for the details.”
Chow rings and decomposition theorems for families of K3 surfaces and Calabi-Yau hypersurfaces (2011, background): “The one used in [2] is that it is enough that x belongs to some (singular) rational curve in S.”
Moduli of sheaves and the Chow group of K3 surfaces (2012, background): “φ M3 i=1 Cp i → 0. (4.0.4) Up to isomorphism the sheaf F is independent of the choice of K 1,K 2,K 3 (notation as in the proof of Proposition 4.2): thus the above construction gives a birational map X[3] 99K Mv(X,H). Let X be a K3 surface and L be an ample line-bundle on X. Let us consider extension”
Chow Rings, Decomposition of the Diagonal, and the Topology of Families (2014, methodology): “Theorem (Beauville and Voisin 2004; see Theorem 5.3).”
On the Chow ring of certain algebraic hyper-K\ (2006, background): “Using the results of [5], this proposition is a consequence of the following lemma:”
Curves and cycles on K3 surfaces (2013, background): “Their ﬁxed point loci can be described, which often contains apart from isolated ﬁxed points and rational curves also smooth elliptic curves and even smooth curves of higher genus, see e.g. [AST11].”
THE MOTIVE OF THE HILBERT CUBE $X^{[3]}$ (2015, methodology): “The Hilbert scheme X[3] of length-3 subschemes of a smooth projective variety X is known to be smooth and projective.”
Remarks And Questions On Coisotropic Subvarieties and 0-Cycles of Hyper-Kähler Varieties (2015, background): “Let now S be a projective K3 surface and let oS ∈ CH0(S) be the canonical 0-cycle of degree 1 on S which is introduced in [4].”
On the Chow ring of birational irreducible symplectic varieties (2013, background): “For K3 surfaces S the subalgebras DCH(S) and R(S) coincide, and the conjectures are known to be true in this case (see [BV04]).”
BLOCH'S CONJECTURE FOR CATANESE AND BARLOW SURFACES (2012, background): “where ∆S is the diagonal of S, oS is the canonical 0-cycle of degree 1 on S introduced in [6], the Ci form a basis of (PicS)⊗Q = NS(S)Q and the αij are the coefficients of the inverse of the matrix of the intersection form of S restricted to NS(S).”
Motivic hyper-Kähler resolution conjecture, I : Generalized Kummer varieties (2016, background): “As an attempt to unify his work on algebraic cycles on abelian varieties [8] and his result with Voisin on Chow rings of K3 surfaces [11], Beauville conjectured in [10], under the name of the splitting property, that for a smooth projective holomorphic symplectic variety X, there exists a canonical ”