Numerical boundedness on rational equivalences of zero cycles on algebraic varieties with trivial \(\mathrm{CH}_0\) (Q1654927)

Let \(X\) be a smooth projective variety over an algebraically closed field of characteristic 0 which is uncountable and \(Z_0X\) denote the group of 0-dimensional algebraic cycles of \(X\). For a zero-cycle \(\alpha\) which is rationally equivalent to zero, a rational equivalence datum for \(\alpha\) is defined to be a finite set \(\{(C,\phi_i)\}\) where \(C_i\)'s are curves on \(X\) and \(\phi_i\)'s rational functions on \(C_i\) such that \(\alpha =\sum\text{div}(\phi_i)\). To a rational equivalence datum, one associates the triple \((m,p_a,d)\) where \(m\) is the number of curves of \(\{C_i\}\), \(p_a\) the maximum of the arithmetic genera \(p_(C_i)\), and \(d\) the maximum of the degrees \(\text{deg}(\phi_i)\). One of the main results of the present paper shows the following: Assume that \(\text{CH}_0(X)\cong\mathbb{Z}\). Then there exists a uniform bound \((m,p_a,d)\) of rational equivalence data for all \(p-q\in Z_0X\) where \(p,q\) go through all pairs of closed points of \(X\). As an application they prove that finite dimensionality of the zero dimensional Chow groups are preserved by degeneration.