Derived categories of Artin-Mumford double solids. (English) Zbl 1475.14033

Hosono, Shinobu; Takagi, Hiromichi

Kyoto J. Math. 60, No. 1, 107-177 (2020).

There are at least two interesting varieties associated to a web of quadrics \(W\): the Reye congruence \(X\), which is an Enriques surface, and the Artin-Mumford quartic double solid \(Y\), which is a Fano 3-fold, famous for being unirational, but not rational.
Let \(V=\mathbb C^4\) and let \(\mathbb P^9\cong \mathbb P(S^2V^*)\) be the space of quadric surfaces. A web of quadrics is a three-dimensional linear subspace \(\mathbb P^3\cong W\subset \mathbb P(S^2V^*)\) satisfying some genericity assumptions; for example, we assume that \(W\) does not contain any quadrics of rank \(1\).
The Artin-Mumford quartic double solid \(Y\) is then defined as the double cover \(Y\to W\) branched over the locus of singular quadrics, which is a quartic in \(W\cong \mathbb P^3\). Note that the variety parametrising lines in a given quadric surface \(Q\) has two connected components if \(Q\) is smooth, but only one component if \(Q\) is singular. Hence, one can regard \(Y\) as the space of pairs \((K,Q)\) consisting of a quadric \(Q\) in \(W\) and a component \(K\) of the variety of lines on \(Q\). The Artin-Mumford quartic double solid is smooth except for 10 ordinary double points. We denote the resolution given by blowing up these points by \(\widetilde Y\to Y\).
The Reye congruence \(X\subset \operatorname*{Gr}(2,V)\) is the locus of lines in \(\mathbb P(V)\) which are contained in at least two quadrics of \(W\).
The varieties \(X\) and \(Y\) are related by the correspondence \[ Z:=\bigl\{(\ell,Q)\mid \ell\subset Q\bigr\}\subset \operatorname*{Gr}(2,V)\times W\,. \] Indeed, there are maps \(Z\to \operatorname*{Gr}(2,V)\), \((\ell,Q)\mapsto \ell\) and \(Z\to Y\), \((\ell,Q)\mapsto(K_\ell,Q)\) where \(K_\ell\) is the component of the variety of lines on \(Q\) containing \([\ell]\).
When mathematicians interested in derived categories recognize two interesting varieties related by a correspondence, they often try to relate the derived categories of coherent sheaves using Fourier-Mukai functors associated to the correspondence.
A first indication that there might indeed be an strong relation between \(\mathcal D^b(\widetilde Y)\) and \(\mathcal D^b(X)\) was that both contain completely orthogonal exceptional collections of length 10. On \(\widetilde Y\), this exceptional collection is given by the structure sheaves of the fibres over the double points of \(Y\), while \(X\) has such a collection consisting of certain line bundles; see [S.Zube, Math.Notes 61, No.6, 693–699 (1997; Zbl 0933.14023)].
Going much further, Ingalls and Kuznetsov [Math.Ann.361, No. 1–2, 107–133 (2015; Zbl 1408.14069)] described full semi-orthogonal decompositions of \(\mathcal D^b(X)\) and \(\mathcal D^b(\widetilde Y)\), such that all components of the decomposition of \(\mathcal D^b(X)\) also show up in the decomposition of \(\mathcal D^b(\widetilde Y)\). This led them to conjecture that \(\mathcal D^b(X)\) embedds as a whole into \(\mathcal D^b(\widetilde Y)\), which means that there should be a fully faithful Fourier–Mukai functor \[ \mathcal D^b(X)\hookrightarrow \mathcal D^b(\widetilde Y)\,. \] In the paper under review, the authors prove this conjecture under a further genericity assumption on \(W\), namely that the locus of singular quadrics in \(W\) does not contain a line.
As one would expect, the Fourier-Mukai kernel of the fully faithful embedding \(\mathcal D^b(X)\hookrightarrow\mathcal D^b(\widetilde Y)\) is related to the correspondence \(Z\), or rather its pull back \(\widetilde Z=Z\times_Y \widetilde Y\). However, the construction is much more involved than just taking the structure or ideal sheaf. Instead, it is a family of rank 2 reflective sheaves on \(\widetilde Y\) parametrized by \(X\).

Reviewer: Andreas Krug (Hannover)

MSC:

14F08	Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
14J45	Fano varieties
14J28	\(K3\) surfaces and Enriques surfaces

Keywords:

Artin-Mumford double solid; Enriques surface; Reye congruence; homological projective duality

Citations:

Zbl 0933.14023; Zbl 1408.14069

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