3. Rational curves on quartics in $P^3$
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1 3 Rational curves on quartics in $P^3$ Objekttyp Chapter Zeitschrift L'Enseignement Mathématique Band (ahr) 43 (997) Heft 3-4 L'ENSEIGNEMENT MATHÉMATIQUE PDF erstellt am Nutzungsbedingungen Die ETH-Bibliothek ist Anbieterin der digitalisierten Zeitschriften Sie besitzt keine Urheberrechte an den Inhalten der Zeitschriften Die Rechte liegen in der Regel bei den Herausgebern Die auf der Plattform e-periodica veröffentlichten Dokumente stehen für nicht-kommerzielle Zwecke in Lehre und Forschung sowie für die private Nutzung frei zur Verfügung Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und den korrekten Herkunftsbezeichnungen weitergegeben werden D Veröffentlichen von Bildern in Print- und Online-Publikationen ist nur mit vorheriger Genehmigung der Rechteinhaber erlaubt Die systematische Speicherung von Teilen des elektronischen Angebots auf anderen Servern bedarf ebenfalls des schriftlichen Einverständnisses der Rechteinhaber Haftungsausschluss Alle Angaben erfolgen ohne Gewähr für Vollständigkeit oder Richtigkeit Es wird keine Haftung übernommen für Schäden durch die Verwendung von Informationen aus diesem Online-Angebot oder durch d Fehlen von Informationen Dies gilt auch für Inhalte Dritter, die über dieses Angebot zugänglich sind Ein Dienst der ETH-Bibliothek ETH Zürich, Rämistrse 0, 8092 Zürich, Schweiz, wwwlibraryethzch http//wwwe-periodicach
2 MV parametrizing ail rational, reduced and irreducible curves belonging to \X\\ and having only nodes for singularities We define W\ the union thèse varieties WY D Remarks ) The schemes parametrizing ail curves a given géométrie genus, in a linear System on a rational surface, hâve been well examined (see [Ta], [Ha]) The new feature in Lemma 26 is that we "pull them up" to subschemes the Chow variety lz m 2 with six points 2) We can think a smooth cubic surface being blown up Then, if we consider the effect blowing-down on the curves P2P the linear system \X\\, we see that Lemma 25 h the following interesting conséquence in the System plane curves degree 3À with six À-fold points, there are some rational, reduced and irreducible curves with only nodes further singularities 3 Lp Rational curves on quartics in $P^3$ A rational space curve degree 8 is given the image a map P3, defined by four homogeneous polynomials degree 8 Such P > maps dépend on 4-9 = 36 arbitrary coefficients; hence they are parametrized 35 l by Those maps which are generically injective and for which (p(p ) is 35 8 a curve degree correspond to an open subset UC By (p GUwe P35P P35P mean that the coefficients Lp are in U Such a curve F is contained in at most one quadric convenient to consider the pair (F, Q) instead F alone shall restrict to the ce where Q is smooth We dénote and Ck) the qui-projective variety smooth quadrics 3 quartics) in P Q So, it will be For simplicity, we by Cq (resp, C c (resp, cubics and LEMMA 3 The following correspondences between qui-projective varieties are algebraic and define closed subvarieties a) the incidence correspondence G C IZ% x Cq parametrizing the rational curves degree 8 on smooth quadrics ; b) the incidence correspondence T C Ti% x Cq x Ck parametrizing the rational, reduced and irreducible curves type (4, 4) on smooth quadrics which are eut out by smooth quartic surfaces; c) the incidence correspondence H C 7Z% x Cq x Tk parametrizing the rational, reduced and irreducible curves type (4,4) on smooth quadrics
3 Pro Let gn# means This is a closed subset UxP3 xcq x * Indeed, (^(P ) C So, the coefficients ail that <p(u for every (m monomials in (u f) fo(p(u 0) and fx^-piii r)) must vanish (hère Q and jfc dénote the polynomial équations g and K) This yields algebraic relations between x and the coefficients (p, fq, and /# 0G r)ggnk PlP f Let Fj be any irreducible component F Call Uj its first projection By [H-P] (Chap, 6, Thm II), there exists an irreducible correspondence Ti between Cq x Ck and an irreducible subvariety lzjji 7Z% which defines the same curves Uj We define T to be the union the T\ Similarly, we define G = {(ipx g) GUxP3 xcq \x G (^(P ) C g} Further, let ((f,x, Q K) GUxP3x Q x^ xg (^(P ) Cgn As before, G and // are closed subsets U x 3 x Cq, respectively U x 3 x Cq x ^ Making use irreducible components G/ G, respectively Hf //, we find irreducible correspondences Qi and 7Y/ between Cq, respectively Cq x 7^ and irreducible varieties!zwt C 7^B, respectively T^v, C 7^B- Again, we define G and H the unions thèse irreducible H={ components Finally, we note that T', G, and varieties, are qui-projective H, closed subsets qui-projective LEMMA 32 The incidence correspondence Tisa qui-projective variety dimension 34 Pro Define tt 0 G-> >Cq by 7r 0 (7, Q) =Q (for 7G 7^ 8 ) and similarly H is a closed subset a qui-projective variety, and FKFK is projective It follows from [Sh] (Chap, 5, Thm 3) that tv^h) is closed in G, and Now, hence also qui-projective Since a smooth quadric Q is projectively normal and a linear équivalence cls on Q is determined by the bidegree, every reduced and irreducible curve F bidegree (44) is eut out, on g, by (at let) one irreducible quartic K Let 7 be the Chow point T, so that (jqk) G H Then the fibre above (7 Q) contains is dimension 0 tt { K and ail the reducible quartics through g Hence it
4 We also know from Lemma 22 that a gênerai member the linear system ail quartics through F is smooth Hence n\ (H) n{t) and the fibres over iï{~) are also 0-dimensional Finally, by Lemma 24, ttootti maps onto Cq and ail the fibres ttoltt^) hâve dimension 5 As Cq is irreducible dimension 9, any irreducible 7 ) maximal dimension h dimension 24 Further, the component fibre 7T over tï{t) is 0-dimensional, so T h dimension 34 7T LEMMA 33 The incidence correspondence C 7Z\2 x c x # parametrizing the rational, reduced and irreducible curves which are the complète intersection a smooth quartic and a smooth cubic surface, is a qui-projective variety dimension 34 Pro The argument is much the same for degree 8 For instance, curve F degree 2 cannot be contained in more than one cubic C So, it is convenient to consider the pair (F, C) instead F alone For simplicity, we restrict to the ce where C is smooth Then Lemma 26 replaces Lemma 24, and the pro Lemma 32 h to be modified mainly for the actual computation dimensions, which is follows The qui-projective variety Ce smooth cubic surfaces h dimension 9 And the dimension the family rational curves FG X4 is equal to, by 5 ) Lemma 26 Finally, a curve FCC lying on an irreducible quartic X, is also contained in ail the reducible quartics through C Hence the linear System ail quartics through F h dimension 4, and a gênerai member is smooth Putting everything together, we find = 34 for the dimension the incidence correspondence a We now corne to the pro sertion (S) for h = 2 and h =3 The smooth quartics in 3 carrying rational, reduced and irreducible curves degree 8, respectively 2, obtained intersections with smooth quadrics, respectively cubics, form a constructible set dimension 34 THEOREM 34 in Ck- A gênerai quartic in this set carries also some rational curves degree 8 (resp, 2 j having only no de s for singularities Pro Let p T> Ck be the projection map defined by p(7, Q,K) =K Then every fibre p is finite Indeed, let us consider the first projection 5 ) Strictly speaking, this is only a lower bound, since Lemma 26 does not count the curves having other singularities than nodes But the pro Theorem 34 shows that one h in fact equality
5 from Lemma 3 that Tis algebraic Hence, for any KeC K, the push-forward q*p~ l (K) describes an algebraic System rational since curves on X, which cannot be dimension >, by Lemma, Moreover, is finite T this algebraic System K CK is ak3 surface Hence do therefore not belong which 8, reduced degree curves parametrizes only to more than one quadric Hence each Chow point 7 corresponds to a unique pair (7,0 Thus the fibre p~\k) contains only finitely many points Let E C T be an irreducible component top dimension 34 By the theorem on the dimension the fibres (see [Sh], Chap, 6, Thm 7), we see that dim p(e) = dime = 34 On applying Chevalley's theorem to the qui-projective varieties T and is constructible, K and to the finite-type morphism p, we also see that p(t) Le, a finite disjoint union locally closed subsets Vt Since K is qui projective,so are the Vf The same argument works for curves degree 2, with the map maximal Ck Note that one gets, for a component E C p q jt > 7^ 8> We know CXC CKC > dimension, Hence we obtain the same equality Remarks ) As expected, before D singular points other than nodes do not affect the relevant schemes This is because, roughly speaking, nodes impose the lowest number conditions for decreing the géométrie genus However, is shown by Lemma 2, not ail curves in Theorem 34 hâve only nodes for singularities 2) In the pro Theorem 34, we could replace E by its closure E in 7^-8 xfqx >k, where Tq dénotes the space ail quadrics in P 3 Now, 7^B x Tq is a projective variety Hence p(ë) is closed (cf [Sh], Chap, 5, Thm 3) and p(e) = Ck- This would account in particular for the rational octics that lie on a quadratic cône, instead a smooth quadric surface the dimensions 4 Rational curves on K3 surfaces in $P^4$ 4 be a K3 surface spanning (Le, not contained in any The notation refers the fact that such a surface is a smooth hyperplane) to complète intersection a quadric and a cubic threefold We also write 2 3 for the 43-dimensional qui-projective variety ail 523 's in 4 (see Lemma 42) In the présent section we prove Let 5*2,3 P4P S2S P4P
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