C VOLL APPENDIX BY A BEAUVILLE GAFA

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292 C. VOLL (APPENDIX BY A. BEAUVILLE) GAFA Appendix: Lines on Pfaffian Hypersurfaces by A. Beauville The aim of this appendix is to prove that a general pfaffian hypersurface of degree r > 2n ? 3 in Pn contains no lines (Proposition 1). By a simple dimension count (see Corollary 4 below), it suffices to show that the variety of lines contained in the universal pfaffian hypersurface (that is, the hyper- surface of degenerate forms in the space of all skew-symmetric forms on a given vector space) has the expected dimension. We will deduce this from an explicit description of the pencils of degenerate skew-symmetric forms, which is the content of the proposition below. We work over an algebraically closed field k. We will need an elementary lemma: Lemma 4. Given a pencil of skew-symmetric forms on a n-dimensional vector space, there exists a subspace of dimension [n+1 2 ] which is isotropic for all forms of the pencil. Proof. By induction on n, the cases n = 0 and n = 1 being trivial. Let ?+ t? be our pencil; we can assume that ? is degenerate. Let D be a line contained in the kernel of ?, and let D? be its orthogonal with respect to ?. Then ? and ? induce skew-symmetric forms ?¯ and ?¯ on D?/D; by the induction hypothesis there exists a subspace of dimension [n?1 2 ] in D?/D which is isotropic for ?¯ and

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292 C.VOLL (APPENDIX BY A. BEAUVILLE)GAFA Appendix: Lineson Pfaﬃan Hypersurfaces by A. Beauville The aim of this appendix is to prove that a general pfaﬃan hypersurface n of degreer >2n3 inPcontains no lines (Proposition 1). By a simple dimension count (see Corollary 4 below), it suﬃces to show that the variety of lines contained in the universal pfaﬃan hypersurface (that is, the hyper-surface of degenerate forms in the space of all skew-symmetric forms on a given vector space) has the expected dimension. We will deduce this from an explicit description of the pencils of degenerate skew-symmetric forms, which is the content of the proposition below. We work over an algebraically closed eldk. We will need an elementary lemma: Lemma 4.Given a pencil of skew-symmetric forms on an-dimensional ! " n+1 vector space, there exists a subspace of dimensionwhich is isotropic 2 for all forms of the pencil. Proof. By induction onn, the casesn= 0 andn= 1 being trivial. Let ϕ+tψbe our pencil; we can assume thatϕis degenerate. LetDbe a line ⊥ contained in the kernel ofϕ, and letDbe its orthogonal with respect ⊥ ¯ toψ. Thenϕandψinduce skew-symmetric forms¯ϕandψonD /D; ! " n1 by the induction hypothesis there exists a subspace of dimensionin 2 ¯ ⊥ ⊥ D /Dwhich is isotropic for¯ϕandψ. The pull-back of this subspace inD ! " n+1 has dimensionand is isotropic forϕandψ. 2 The following result must be well known, but I have not been able to nda reference: 1 Proposition 5.LetVbe a vector space of dimension2r, and(ϕt)∈Pa t pencil ofdegenerateskew-symmetric forms onV. There exists a subspace 1 L⊂Vof dimensionr+ 1which is isotropic forϕtfor allt∈P. Proof. Again we prove the proposition by induction onr, the caser= 1 be-∗ ing trivial. The associated maps Φt:V→Vform a pencil of singular linear maps. By a classical result in linear algebra (see [G, Chap.XII, Thm.4]),  ∗ there exist subspacesK∈VandL∈V, with dimK= dimL+ 1, such  that Φt(K)⊂Lfor allt; equivalently, there exist subspacesKandLofV, with dimK+ dimL= 2r+ 1, which are orthogonal for eachϕt. Replacing (K, L) by (K∩L, K+L) we may assumeK⊂L; the pencil (ϕt) restricted toLis singular onK, hence induces a pencil (¯ϕt) onL/K. Put dimK=p, so that dim(L/K) = 2r+ 12p. By the above lemma there is a subspace

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