Exceptional Collections and Twisted Cubics on Fano Varieties

Definitions and mathematical statement

Let $X$ denote the Küchle fourfold of type $(c5)$ in the supplied record, and let $Y$ denote its associated fivefold. The record describes $X$ as a half-anticanonical section of $Y$. In the usual smooth projective interpretation, this means that there is a line bundle $L$ on $Y$ such that \[ L^{\otimes 2}\cong\omega_Y^{-1},\qquad X\in |L|. \] Here $\omega_Y$ is the canonical line bundle and $|L|$ is the complete linear system of $L$. The supplied record does not give the construction of $Y$, the ground field, the required smoothness or generality hypotheses, or the precise family quantifier. These data must be recovered from the cited report before this becomes a fully specified statement.

For a smooth projective variety $Y$ over a field $k$, write $\mathrm D^b(\operatorname{Coh}Y)$ for the bounded derived category of coherent sheaves. An object $E$ is exceptional if \[ \operatorname{Hom}(E,E[p])\cong \begin{cases} k,&p=0,\\ 0,&p\ne 0. \end{cases} \] An ordered collection $(E_1,\ldots,E_N)$ is exceptional if every $E_i$ is exceptional and $\operatorname{Hom}(E_j,E_i[p])=0$ for all $j>i$ and all $p\in{\mathbb{Z}}$. It is full if its objects generate $\mathrm D^b(\operatorname{Coh}Y)$ as a triangulated category closed under direct summands.

A rectangular Lefschetz exceptional collection with respect to a specified ample line bundle $M$ consists of repeated blocks \[ \begin{split} (&E_1,\ldots,E_m,\ E_1\otimes M,\ldots,E_m\otimes M,\ \ldots,\\ &E_1\otimes M^{\otimes(r-1)},\ldots,E_m\otimes M^{\otimes(r-1)}), \end{split} \] where $m,r$ are positive integers and the displayed ordered collection is full and exceptional. Equivalently, with $\mathcal A=\langle E_1,\ldots,E_m\rangle$, it gives the semiorthogonal decomposition \[ \mathrm D^b(\operatorname{Coh}Y) =\langle\mathcal A,\mathcal A\otimes M,\ldots, \mathcal A\otimes M^{\otimes(r-1)}\rangle. \] The first supplied conjectural assertion is that the associated $Y$ admits such a collection with \[ mr=12. \] The record does not specify $M$, identify it explicitly with the half-anticanonical line bundle $L$, or specify the number and size of the blocks. Thus this describes the required structure but does not resolve those missing parameters.

To interpret the second assertion, fix a projective embedding of $X$ with very ample line bundle $A$. A smooth rational twisted cubic means a smooth curve which, after passage to an algebraic closure, is a rational normal cubic in its projective linear span ${\mathbb{P}}^3$. Its Hilbert polynomial with respect to $A$ is \[ P_C(t)=\chi(C,A^{\otimes t}|_C)=3t+1. \] The scheme $\operatorname{Hilb}^{3t+1}(X;A)$ parametrizes closed subschemes with this Hilbert polynomial, but need not consist entirely of smooth rational twisted cubics. Let $\mathscr H$ stand for the particular locus, closure, or irreducible component intended by the report's phrase "Hilbert scheme of rational twisted cubic curves on $X$." This notation records an unresolved choice; the supplied record does not identify that parameter space or the embedding $A$.

The second supplied conjectural assertion is the existence of a hyperkähler fourfold $Z$ and a birational equivalence \[ \mathscr H\dashrightarrow Z. \] In the standard complex algebraic interpretation, a projective irreducible holomorphic symplectic fourfold is a smooth simply connected projective complex variety $Z$ of complex dimension $4$ such that \[ H^0(Z,\Omega_Z^2)={\mathbb{C}}\sigma \] for an everywhere nondegenerate holomorphic two-form $\sigma$. The source's precise convention for "hyperkähler," including projectivity, is not supplied. The displayed birational assertion requires a specified integral parameter space $\mathscr H$; if the relevant Hilbert scheme is reducible, its intended component must first be identified.

Both assertions are recorded as conjectural targets. Their intended scope within the family of pairs $(Y,X)$, including whether they concern every smooth member, a general member, or a very general member, is not specified in the supplied text.

Short English statement

For the associated fivefold $Y$ and Küchle fourfold $X$ of type $(c5)$, the source asks for a full rectangular Lefschetz exceptional collection of twelve objects on $Y$, and for the intended parameter space of rational twisted cubics on $X$ to be birational to a hyperkähler fourfold. The precise geometric construction, family quantifiers, Lefschetz polarization, and curve parameter space require clarification from the cited report.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002827$, title "Exceptional Collections and Twisted Cubics on Fano Varieties," exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The proposed year is recorded as 2015; no individual proposer is identified.
• Supplied primary report citation: Algebraic Geometry (2015), Oberwolfach Reports, DOI 10.4171/owr/2015/15, pp. 783--836. Supplied literature-source label: "OWR Report 15/2015." https://doi.org/10.4171/owr/2015/15.
• Supplied literature reference: Kuznetsov, "Kuechle fivefolds." https://arxiv.org/abs/1603.03161.
• The supplied literature assessment was checked on 2026-08-21 and labels the problem "partially solved." It reports that later work supplies birational and motivic information for the relevant Fano fourfold, while the asserted rectangular length-twelve exceptional collection and twisted-cubic Hilbert-scheme birationality remain conjectural. This is supplied dated curation, explicitly described in the record as literature triage requiring independent verification; neither its status assessment nor its cited claims have been independently verified here.
• The registry labels the statement corrected_verified and says that it was checked against the official Oberwolfach report and preserves that report's hypotheses, notation, and target. This is a supplied verification claim, not verification performed for this draft. The supplied original wording refers to "the fivefold $Y$ and associated fourfold $X$ in the report," without reproducing their construction or an exact internal report location.