Foliations by curves on threefolds

Abstract

We study the conormal sheaves and singular schemes of 1-dimensional foliations on smooth projective varieties X of dimension 3 of Picard rank 1. We prove that if the singular scheme has dimension 0, then the conormal sheaf is µ-stable whenever the tangent bundle T X is stable, and apply this fact to the characterization of certain irreducible components of the moduli space of rank 2 reflexive sheaves on P 3 and on a smooth quadric hypersurface Q 3 ⊂ P 4 . Finally, we give a classification of local complete intersection foliations, that is, foliations with locally free conormal sheaves, of degree 0 and 1 on Q 3 . Contents17 7.3. Foliations of odd degree 19 7.4. Local complete intersection foliations of degree 0 20 7.5. Local complete intersection foliations of degree 1 21 References 22

Codimension one distributions of degree 2 on the three-dimensional projective space

“…Dualizing the second column from the left we get a foliation by curves In particular, taking d = 2 in the statement above implies that there exists a codimension one distribution D of degree 2 such that (c 2 (T D ), c 3 (T D )) = (5,14). Remark 34.…”

“…Since every one-dimensional distribution is automatically integrable, we call (6) a foliation by curves of degree k on P 3 . For a detailed account on foliations by curves we refer to [5,10]. We will be mostly interested in the cases k = 0 and k = 1.…”

“…We recall that the spectrum of a rank 2 reflexive sheaf is a set of integers that encodes some partial information on the cohomology of F , see Section 2.4. Except for the cases (5,14), (4,8) and (4,6), the spectrum that appears in the table is the only one that is realized by codimension one distributions; regarding these three special cases, see Remarks 31 and 35.…”

“…The first technique, described in Section 3.1, relies on codimension two distributions on P 3 , i.e., foliations by curves: given a codimension two distribution, one can find a codimension one distribution containing it and whose singular locus can be explicitly described, see Proposition 6. This result is applied to prove the existence of codimension one distributions D of degree 2 such that Sing 1 (D) is a pair of disjoint lines, a smooth conic or a line, see Sections 7.2, 7.3 and 8, establishing the existence of the cases (c 2 (T D ), c 3 (T D )) = (4, 8), (4, 10) and (5,14), respectively. It is also interesting to note that our results can be regarded as a proof of existence of stable rank 2 reflexives sheaves with the spectra given in Table 1.…”

Sheaves and distributions on threefold hypersurfaces

“…Techniques from algebraic geometry have been extremely useful in the study of distributions and foliations on complex projective spaces, see for instance [1,2,7,22,23,25,26]. From the point of view of algebraic geometry, a foliation by curves F on a smooth projective threefold X is a short exact sequence of the form The non negative integer r above is called the degree of F .…”