TY - JOUR
T1 - Trisecant lemma for nonequidimensional varieties
AU - Kaminski, Jeremy Yirmeyahu
AU - Kanel-Belov, Alexei
AU - Teicher, Mina
PY - 2008/2
Y1 - 2008/2
N2 - Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For r ≥ 3, if every (r-2)-plane x 1,...,xr-1, where the x i are generic points, also meets X in a point x r different from x 1,..., x r-1, then X is contained in a linear subspace L such that codim L X ≤ r - 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℘r, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℘n + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℘r, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of {l ∈ script G sign(1,r)|∃ p ∈ Y, q1, q2 ∈ Z\Y, q 1, q2,p ∈ l}. We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.
AB - Let X be an irreducible projective variety over an algebraically closed field of characteristic zero. For r ≥ 3, if every (r-2)-plane x 1,...,xr-1, where the x i are generic points, also meets X in a point x r different from x 1,..., x r-1, then X is contained in a linear subspace L such that codim L X ≤ r - 2. In this paper, our purpose is to present another derivation of this result for r = 3 and then to introduce a generalization to nonequidimensional varieties. For the sake of clarity, we shall reformulate our problem as follows. Let Z be an equidimensional variety (maybe singular and/or reducible) of dimension n, other than a linear space, embedded into ℘r, where r ≥ n + 1. The variety of trisecant lines of Z, say V 1,3(Z), has dimension strictly less than 2n, unless Z is included in an (n + 1)-dimensional linear space and has degree at least 3, in which case dim V 1,3(Z) = 2n. This also implies that if dim V 1,3(Z) = 2n, then Z can be embedded in ℘n + 1. Then we inquire the more general case, where Z is not required to be equidimensional. In that case, let Z be a possibly singular variety of dimension n, which may be neither irreducible nor equidimensional, embedded into ℘r, where r ≥ n + 1, and let Y be a proper subvariety of dimension k ≥ 1. Consider now S being a component of maximal dimension of the closure of {l ∈ script G sign(1,r)|∃ p ∈ Y, q1, q2 ∈ Z\Y, q 1, q2,p ∈ l}. We show that S has dimension strictly less than n + k, unless the union of lines in S has dimension n + 1, in which case dim S = n + k. In the latter case, if the dimension of the space is strictly greater than n + 1, then the union of lines in S cannot cover the whole space. This is the main result of our paper. We also introduce some examples showing that our bound is strict.
UR - http://www.scopus.com/inward/record.url?scp=38549176136&partnerID=8YFLogxK
U2 - 10.1007/s10958-008-0047-7
DO - 10.1007/s10958-008-0047-7
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AN - SCOPUS:38549176136
SN - 1072-3374
VL - 149
SP - 1087
EP - 1097
JO - Journal of Mathematical Sciences
JF - Journal of Mathematical Sciences
IS - 2
ER -