Classification of maximal-rank webs at the exceptional value $d=(r+2)(n-1)+1$ with $n\geq 3$, and the existence of non-grassmannian maximal-rank webs below the extremal range $q(d)\geq n-1$

In web geometry, a $d$-web of codimension $r$ on a manifold of dimension $rn$ (a web of type $(r,n)$) is a collection of $d$ foliations of codimension $r$ in general position, and its rank is the dimension of the vector space of its abelian relations, the web-theoretic analogue of the linear relations among holomorphic differentials on an algebraic curve that appear in Abel's theorem. Chern's 1935 paper 'Abzählungen für Gewebe' determined the sharp upper bound for the rank of webs of hypersurfaces in arbitrary ambient dimension; Chern and Griffiths (1978) extended the sharp bound to arbitrary codimension, and the algebraization of maximal-rank webs was later completed outside a single exceptional range (Trépreau 2006 for codimension one, Pirio-Trépreau 2015 for codimension at least two). The open core stated in the modern literature is: determine, up to local isomorphism, all maximal-rank $d$-webs of type $(r,n)$ at the exceptional value $d=(r+2)(n-1)+1$ with $n\geq 3$, where grassmannian algebraization is unavailable and non-grassmannian examples are established only for $n=3$, $n=4$ and for $(r,n)=(2,5)$, $(2,6)$; and, below the extremal range of the algebraization theorem - that is for $0\leq q(d)<n-1$, equivalently $r(n-1)+2\leq d\leq(r+1)(n-1)+1$, the range excluded from the theorem's hypothesis - determine whether there exist maximal-rank $d$-webs that are not isomorphic to a germ of a grassmannian algebraic web beyond the codimension-two webs with $q(d)=0$, the only known counterexamples.

Background

Web geometry (German: Gewebegeometrie) studies configurations of foliations in general position and the functional identities among their first integrals; it originated with the Blaschke school in Hamburg in the 1930s and connects the inverse of Abel's theorem for algebraic curves with local differential geometry and projective algebraic geometry.

A foliation of type $(r,n)$ is a regular foliation $F$ of codimension $r$ on a germ $(M,x_0)$ of a smooth complex analytic manifold of dimension $rn$. If $F$ is locally defined by a system $du_1=\dots=du_r=0$ with linearly independent differentials, a normal of $F$ is an $r$-form $\varphi=f\,du_1\wedge\dots\wedge du_r$ for a function $f$; it is closed when $d\varphi=0$. A family $T=\{F_1,\dots,F_d\}$ of such foliations is a $d$-web of type $(r,n)$ when the generating normals of the $F_j$ are in general position. An abelian relation of $T$ is a $d$-tuple of closed normals $\varphi_j$, one for each foliation, whose sum vanishes identically on $(M,x_0)$; the rank of the web is the dimension of the complex vector space of its abelian relations. This is the exact web-theoretic analogue of Abel's theorem: for the web cut out by the branches of an algebraic curve through a general point of the plane, the abelian relations reproduce the linear relations among holomorphic differentials on the curve.

A basic finiteness result states that the rank of any $d$-web of type $(r,n)$ is at most

$$\rho_{r,n}(d)=\sum_{h\geq 0}\binom{r-1+h}{r-1}\,\max\bigl(d-(r+h)(n-1)-1,\,0\bigr),$$

and that this bound is optimal. The bound was obtained for planar webs ($r=1$, $n=2$) by Bol in 1932, for codimension-one webs in arbitrary dimension by Chern in 1935, and in full generality by Chern and Griffiths in 1978. A web whose rank equals $\rho_{r,n}(d)$ is said to be of maximal rank (in the 1930s literature, of maximal Höchstzahl). Writing $q(d)=d-r(n-1)-2$, the bound is nonzero only for $q(d)\geq 0$, and extremal constructions exist in every admissible case.

The central structural question for maximal-rank webs is their algebraization. On the grassmannian $\mathrm{Gr}_{r,n}$ of $(n-1)$-planes in $\mathbb{P}^{r+n-1}$ (a manifold of dimension $rn$), the Schubert cycles of $(n-1)$-planes passing through a fixed point - called the distinguished subvarieties - are smooth of codimension $r$; a foliation of type $(r,n)$ is grassmannian when its leaves are open pieces of distinguished subvarieties. If $Z\subset\mathbb{P}^{r+n-1}$ is a projective variety of dimension $r$ and degree $d$ such that a general $(n-1)$-plane meets $Z$ in $d$ points in general position, projective duality produces from $Z$ a $d$-web $T_Z$ of codimension $r$ on $\mathrm{Gr}_{r,n}$, called a grassmannian algebraic web. A general $d$-web is algebraizable when it is locally isomorphic to a germ of an algebraic web. Bol proved (for type $(1,3)$, $d\geq 6$) that maximal-rank codimension-one webs are algebraizable; Chern and Griffiths attacked the generalization to all $n\geq 3$, but an error in the third part of their 1978 memoir, acknowledged in their 1981 corrections, forced a considerable weakening of their statement. Trépreau settled the codimension-one case in 2006, and Pirio-Trépreau settled codimension $r\geq 2$ in 2015: a maximal-rank $d$-web of type $(r,n)$ with $q(d)\geq n-1$ is isomorphic to a germ of a grassmannian algebraic web whenever $n=2$ or $q(d)\neq 2n-3$. The bound $\rho_{r,n}(d)$ also gives, via the general Abel theorem of Griffiths, a sharp bound for the geometric genus of the projective varieties $Z$ entering the construction, generalizing Castelnuovo's classical genus bound for curves in projective space.

At the exceptional value $q(d)=2n-3$, that is $d=(r+2)(n-1)+1$ with $n\geq 3$, the Pirio-Trépreau theorem still yields algebraization, but only in a wider class: a germ of an algebraic web, possibly non-grassmannian. There the determination of the maximal-rank webs themselves is explicitly left open: it is unknown, for a given pair $(r,n)$, whether a maximal-rank $d$-web with $q(d)=2n-3$ exists that is not isomorphic to a germ of a grassmannian algebraic web (such webs are known to exist for $n=3$ or $n=4$ with any $r\geq 2$, and for $r=2$ with $n=5$ or $n=6$), and no classification of them is known. Likewise, little is known below the extremal range $q(d)\geq n-1$ of that theorem: the cases $r(n-1)+1<d\leq(r+1)(n-1)+1$ excluded from its hypothesis have not been studied; the canonical method gives no interesting result when $q(d)<n-1$, and whether the grassmannian conclusion fails for maximal-rank webs with $0\leq q(d)<n-1$ is open in general, the only known counterexamples being codimension-two webs with $q(d)=0$.

Problem Statement

Fix integers $r\geq 2$ and $n\geq 3$, and recall $q(d)=d-r(n-1)-2$ and the sharp Chern-Griffiths bound

$$\rho_{r,n}(d)=\sum_{h\geq 0}\binom{r-1+h}{r-1}\,\max\bigl(d-(r+h)(n-1)-1,\,0\bigr)$$

for the rank of any $d$-web of type $(r,n)$. The open problem stated in the literature has two linked parts.

(1) Exceptional maximal-rank webs. Determine, up to local isomorphism of webs, all $d$-webs of type $(r,n)$, $n\geq 3$, of maximal rank $\rho_{r,n}(d)$ at the exceptional value $q(d)=2n-3$, that is $d=(r+2)(n-1)+1$. This includes: (i) for every pair $(r,n)$ with $r\geq 2$ and $n\geq 3$, deciding whether there exists such a web that is not isomorphic to a germ of a grassmannian algebraic web - non-grassmannian examples are known to exist for $n=3$ or $n=4$ with any $r\geq 2$ and for $(r,n)=(2,5)$ and $(2,6)$, and the question is open for every other pair; (ii) classifying all such webs, including the grassmannian ones, in terms of the associated Blaschke varieties - for a maximal-rank web $T$: the projective variety $X_T\subset\mathbb{P}^N$ associated with $T$ by the canonical method of Blaschke, Bol and Howe, namely the intersection of all projective subvarieties of $\mathbb{P}^N$ containing the normal rational curves of degree $q(d)$ attached to the abelian relations of $T$; it is an invariant of the isomorphism class of $T$, and for $r\geq 2$ it is irreducible of dimension $r+1$ - and the integrability of the almost-grassmannian structures (the $G$-structures modeled on the grassmannian of $(n-1)$-planes in $\mathbb{P}^{r+n-1}$, of which the web's family of normals is a prototype) carried by their admissible sets (for a Blaschke variety $X$ and $q=q(d)$: the $rn$-dimensional variety $\Sigma_X$ parameterizing the rational curves of degree $q$ whose supports pass through $n$ general points of $X$, together with its Zariski-dense open subset $\Sigma_{X,\mathrm{adm}}$ of admissible elements). The existence question (i) is a consequence of the classification (ii), which classifies all such maximal-rank webs.

(2) Maximal-rank webs below the extremal range. Determine, for each pair $(r,n)$ with $r\geq 2$ and $n\geq 3$, whether there exists, in the regime $0\leq q(d)<n-1$ - equivalently $r(n-1)+2\leq d\leq(r+1)(n-1)+1$, the range the hypothesis of the Pirio-Trépreau theorem excludes and which has not been studied - a maximal-rank $d$-web of type $(r,n)$ that is not isomorphic to a germ of a grassmannian algebraic web, beyond the known codimension-two examples with $q(d)=0$. The canonical method gives no interesting result when $q(d)<n-1$, so this regime lies outside the machinery that settled the complementary range.

References

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