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- W4300868986 abstract "We are interested by holomorphic $d$-webs $W$ of codimension one in a complex $n$-dimensional manifold $M$. If they are ordinary, i.e. if they satisfy to some condition of genericity (whose precise definition is recalled), we proved in [CL] that their rank $rho(W)$ is upper-bounded by a certain number $pi'(n,d) bigl($which, for $ngeq 3$, is stictly smaller than the Castelnuovo-Chern's bound $pi(n,d)bigr)$. In fact, denoting by $c(n,h)$ the dimension of the space of homogeneous polynomials of degree $h$ with $n$ unknowns, and by $h_0$ the integer such that $$c(n,h_0-1)<dleq c(n,h_0),$$ $pi'(n,d)$ is just the first number of a decreasing sequence of positive integers $$pi'(n,d)=rho_{h_0-2}geq rho_{h_0-1}geq cdotsgeq rho_{h}geq rho_{h+1}geqcdotsgeq rho_{infty}=rho(W)geq 0 $$ becoming stationary equal to $rho(W)$ after a finite number of steps. This sequence is an interesting invariant of the web, refining the data of the only rank. The method is effective : theoretically, we can compute $rho_h$ for any given $h$ ; and, as soon as two consecutive such numbers are equal ($rho_h=rho_{h+1}, hgeq h_0-2$), we can construct a holomorphic vector bundle $R_hto M$ of rank $rho_h$, equipped with a tautological holomorphic connection $nabla^h$ whose curvature $K^h$ vanishes iff the above sequence is stationary from there. Thus, we may stop the process at the first step where the curvature vanishes. Examples will be given." @default.
- W4300868986 created "2022-10-04" @default.
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- W4300868986 date "2017-03-10" @default.
- W4300868986 modified "2023-09-25" @default.
- W4300868986 title "Rank of ordinary webs in codimension one. An effective method" @default.
- W4300868986 doi "https://doi.org/10.48550/arxiv.1703.03725" @default.
- W4300868986 hasPublicationYear "2017" @default.
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