We investigate k-nets with k 4 embedded in the projective plane PG(2;K) defined over a field K; they are line congurations in PG(2;K) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If K has zero characteristic, no embedded k-net for k >=5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.
K-NETS EMBEDDED IN A PROJECTIVE PLANE OVER A FIELD
KORCHMAROS, Gabor;
2014-01-01
Abstract
We investigate k-nets with k 4 embedded in the projective plane PG(2;K) defined over a field K; they are line congurations in PG(2;K) consisting of k pairwise disjoint line-sets, called components, such that any two lines from distinct families are concurrent with exactly one line from each component. The size of each component of a k-net is the same, the order of the k-net. If K has zero characteristic, no embedded k-net for k >=5 exists; see [11,14]. Here we prove that this holds true in positive characteristic p as long as p is sufficiently large compared with the order of the k-net. Our approach, different from that used in [11,14], also provides a new proof in characteristic zero.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
