Chow groups of projective varieties of very small degree
Let \(C\) be a closed subset of the projective space \(\mathbb{P}^n\) over an algebraically closed field. Assume \(X\) is defined by \(r\) equations of degree \(d_1\geq\cdots\geq d_r\). Then one defines the numerical invariant \(k=[(n-\sum^r_{i=2}d_i)/d_1]\), where \([\alpha]\) denotes the integral part of a rational number \(\alpha\). A conjecture formulated by Srinivas and \textit{K. P. Paranjape} [Ann. Math., II. Ser. 139, No. 3, 641-660 (1994; Zbl 0828.14003)] states that \(\text{CH}_l(X)\otimes\mathbb{Q}=\mathbb{Q}\) for any \(l\leq k-1\), where \(\text{CH}_l(X)\) denotes the \(l\)-th Chow group of \(X\), i.e. the group of \(l\)-dimensional algebraic cycles on \(X\) modulo rational equivalence. For zero-cycles the previous conjecture follows from a Roitman theorem. In the quoted paper Paranjape proved a version of the conjecture, showing that there is a finite bound \(N=N(d_1,\dots,d_r;l)\) such that, for \(n\geq N\), one has \(\text{CH}_{l'} (X)= \mathbb{Z}\) for \(0\leq l'\leq l\). In the present paper the authors improve the bound \(N\). More precisely, they show that if \(d_1\geq\cdots\geq d_r\geq 2\) and either \(d_1\geq 3\) or \(r\geq l+1\), and if \(\sum^r_{i=1}{l+d_i\choose l+1}\leq n\), then \(\text{CH}_{l'}(X)\otimes \mathbb{Q}= \mathbb{Q}\) for any \(0\leq l'\leq l\). The same conclusion holds true if \(d_1=\cdots=d_r=2\) and \(r\leq l\), but assuming \(r(l+2) \leq n-l+r-1\). Next, as a consequence, the authors show that if in addition to the above inequality one assumes \(X\) is smooth, then the primitive cohomology of \(X\) is generated by the image of the homology of a \((l+1)\)-codimension subset, in accordance with Grothendieck's conjecture. Moreover they prove that, when \(X\) is defined in characteristic zero, the number of \(\mathbb{F}_q\)-rational points verifies the congruence \(\#\mathbb{P}^n(\mathbb{F}_q) \equiv \#X(\mathbb{F}_q)\pmod q^{l+1}\) for almost all primes \(p\).
scientific article; zbMATH DE number 1011422
scientific article