Computations with modified diagonals (Q467375)

The \(m\)-th modified diagonal on a variety \(X\) over a field \(\mathbb{K}\) is a cycle defined by NEWLINE\[NEWLINE \Gamma^m(X;a) := \sum_{J\not=\emptyset}(-1)^{m-|J|}\Delta^m_J(X;a), NEWLINE\]NEWLINE where for a \(\mathbb{K}\)-rational point \(a\in X(\mathbb{K})\) and a subset \(J\subset \{ 1,\ldots , m\}\) NEWLINENEWLINE\[NEWLINE\Delta^m_J(X;a) := \Big\{ (x_1,\ldots , x_m)\in X^m \, \Big| \, \begin{cases} x_i=x_j\, &\text{if \(i\) and \(j\) are in \(J\)},\\ x_k=a \, &\text{if \(k\) are not in \(J\)}\end{cases} \Big\}. NEWLINE\]NEWLINENEWLINENEWLINEBeing motivated by Beauville's conjecture of the vanishing of the modified diagonal on a hyperkähler variety in the rational Chow group, the article under review states the main results in Propositions 0.2 and 0.3.NEWLINENEWLINEIn Proposition 0.2, for smooth projective varieties \(X\) and \(Y\), if \(\Gamma^m(X;a)\) and \(\Gamma^n(Y;b)\) are both rationally equivalent to zero for some \(a\in X(\mathbb{K})\) and \(b\in Y(\mathbb{K})\), then, \(\Gamma^{m+n-1}(X\times Y;(a,\, b))\) is shown to be rationally equivalent to zero. By applying to smooth genus-2 curves, it is proved that the \(5\)-th modified diagonal of a complex abelian surface is rationally equivalent to zero at some point.NEWLINENEWLINEIn Proposition 0.3, a blow-up \(X\to Y\) centred along a codimension-\(e\) smooth subvariety \(V\) in a smooth projective variety \(Y\) of dimension \(n\) is considered. If for some \(b\in V(\mathbb{K})\), \(\Gamma^{n+1}(Y;b)\) and \(\Gamma^{n-e+1}(V;b)\) are both rationally equivalent to zero, then, it is proved that \(\Gamma^{n+1}(X;a)\) vanishes for some \(a\in f^{-1}(b)\).NEWLINENEWLINEThe Hilbert scheme \(S^{[n]}\) of a \(K3\) surface \(S\) of subvarieties of length \(n\) is constructed by blowing-up \(S^{[n+1]}\to S^{[n]}\times S\) centred along the tautological subscheme \(\mathcal{L}_n\) of \(S^{[n]}\times S\). It was shown by Beauville that \(S^{[n]}\) is hyperkähler, and by Beauville-Voisin that the \(3\)rd modified diagonal of \(S\) vanishes at a point \(c\) on a rational curve on \(S\). An application of Proposition 0.3 shows that there exists \(a_n\in S^{[n]}\) representing a scheme supported at \(c\) such that \(\Gamma^{2n+1}(S^{[n]}; a_n)\) is rationally equivalent to zero for \(n=2,3\), being \(\mathcal{L}_n\) smooth. Note that for general \(n\geq 3\), the tautological subscheme \(\mathcal{L}_n\) is singular so that Proposition 0.3 cannot be applied for general cases.NEWLINENEWLINEIn other parts, vanishing theorems of modified diagonals of an \(\mathbb{P}^r\)-bundle over a smooth projective variety of dimension \(1\) or \(2\), and of a double cover are also discussed.NEWLINENEWLINEThe reviewer wonders if it is possible to prove analogous statements (under appropriate hypotheses) to an \(\mathbb{P}^r\)-bundle over a smooth projective variety of dimension \(\geq 3\), and whether or not Proposition 0.3 extends to the case where the blow-up centre is singular in order to prove for instance a vanishing theorem of the Hilbert scheme \(S^{[n]}\) of a \(K3\) surface \(S\) for all \(n\).