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ON the VARIETY of PLANE CURVES of DEGREE D with Ф transactions of the american mathematical society Volume 316, Number 1, November 1989 ON THE VARIETY OF PLANE CURVES OF DEGREE d WITH Ô NODES AND /c CUSPS PYUNG-LYUN KANG Abstract. Let Pw be the projective space which parametrizes all plane curves of degree d and V(d,S, k) the subvariety of P^ consisting of all reduced and irreducible plane curves of degree d with S nodes and k cusps as their only singularities. In this paper we prove that V(d, S, k) is irreducible if k < 3 , except possibly when k = 3 and d = 5 or 6. 1. Introduction In this paper we study the locus V(d ,S ,k) of reduced and irreducible plane curves of degree d with ô nodes and k cusps as their only singularities. Let P be the projective space parametrizing plane curves of degree d , and Vd the locally closed subset of P of reduced and irreducible plane curves of degree d and geometric genus g . Harris [HI] recently proved that the variety Vd is irreducible. Knowing this fact, Diaz and Harris [DH1 ] showed that ( 1.1 ) If IV c Vd is any subvariety of codimension 1, and D E IV a general point, then the singularities of D are either (l) m nodes, (2) m - 1 nodes and one cusp (CU), (3) m - 2 nodes and one tacnode (TN), or (4) m - 3 nodes and one ordinary triple point (TR), where m — j(d - l)(d - 2) - g. They asked in [DH2] whether those loci are irreducible. In this paper we prove that the variety V(d ,S, 1), CU in (1.1), is irreducible. We essentially follow the same idea as that in [HI]; first we show V(d, j(d— I)(d-2)-1,1), the locus of reduced and irreducible rational plane curves of degree d with \(d - l)(d - 2) - 1 nodes and one cusp, is irreducible; second we show that there exists only one component of V(d,5,1) which contains V(d,{(d-l)(d-2)-l,l) in its closure; finally we prove that every component of V(d,ô, 1) admits a degeneration to V(d, \(d - l)(d - 2) - 1,1). Using the same argument as above we prove the more general theorem. Received by the editors April 4, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H10. Key words and phrases. Plane curves, cusps, families of plane curves with nodes and cusps. ©1989 American Mathematical Society 0002-9947/89 $1.00+ $.25 per page 165 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 166 PYUNG-LYUN KANG ( 1.2) Theorem. The variety V(d, ô, k) is irreducible if k <2. If k = 3, then, except possibly when d = 5 or 6, V(d ,6 ,k) is irreducible. Remark. Ziv Ran has recently and independently shown that V(d,5,l) is irreducible. He has also shown that the two other divisors TN and TR of V.a,g in (1.1) are irreducible. (1.3) Definitions and notations. We mean by the (geometric) genus g(C) of a reduced curve C the genus of its normalization; in particular, if C has irreducible components C, , C7, ... , Ck , then £(C) = X>(C,)-/c + l. i If C is a reduced plane curve, then the class c(C) of C is the degree of the dual curve C* in (P )*, the projective plane of lines of P . If a reduced plane curve C has ô nodes and k cusps as its only singularities, then g(C) = \(d-I)(d-2)-ô-K and c(C) = d(d - 1) - 28 - 3k. By a nice curve with given singular points we mean a curve all other singu- larities of which besides given ones are nodes. Acknowledgment. I would like to thank my thesis advisor Dr. Steven Diaz for his helpful advice and careful proofreading throughout the preparation of this paper. 2 In this section we prove the first two claims before (1.2) in the Introduction. (2.1) Theorem. Assume d > 3. Then V(d, \(d - l)<d - 2) - k ,k) c Vd0 is irreducible if K < min(d - 2, \(d + 1)). Proof. Note that any rational plane curve of degree d can be realized as a projection nA of a rational normal curve C in the projective space P from a (d - 3)-plane A. Fix a rational normal curve C and a projective plane P to project onto once and for all. Let Gr(úí - 3,P ) be the Grassmannian of (d - 3)-planes of Pd and "V the dense subset in Gr(d - 3,Pd) of (d - 3)-planes which occur in the above fact and avoid the fixed projective plane. Then we have a map it from 'V to Vd0 modulo Aut(P ) defined by 7z(A) = nA(C) for Ae'V . It is therefore enough to show that n~x(V(d ,\(d- l)(d - 2) - k , k)) is irreducible in Gr(d - 3, P ). In fact, since any automorphism of C is given by an automorphism of P , we have nA(C) — nA,(C) if and only if A = A (A1) where A lies in Aut(C) c Aut(P ). Let A e n~x(V(d,\(d - l)(d - 2) - k,k)) . So A meets k distinct tangent lines, say, í of C away from C, where t¡ are tangent lines of C at pi e C, for / = 1, ... ,rc. Let o2(t¡) = {(d - 3)-planes meeting tt} which is a codimension two Schubert variety of Gr(d - 3,Pd) (see [GH] for notations). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use PLANE CURVES OF DEGREE d 167 Then A E f|*=1 o2(t¡) and a general (d - 3)-plane in n*=1 a2^¡) meets omy K distinct tangent lines, otherwise (i.e. if it meets more than k distinct tangent lines of C ) it will be a general member of a subvariety with higher codimension (cf. [EH, Theorem (2.3)]). Let CK be the Kth symmetric product of C and A be the diagonal of CK, i.e., A = {(px, ... ,pK) E CK\pj = p-t for some distinct i and /'}. Note that CK\A is irreducible. Since, from what we observed before, n~x(V(d,{(d-l)(d-2)-K,K)) can be considered as a fiber space over CK\A with a fiber over a point (px, ... , pK) in CK\A an open and dense set of f|/=i M'/) consisting of (d - 3)-planes which meet t¡ transversely away from C, it is enough to show that f\*_y a2^¡) is irreducible and of constant dimension. For this purpose we now fix for a while k distinct points px, ... ,pK on C and the tangent lines t¡ of C at pr Let Xj E t¡, i = 1, ... ,k , and a3(x¡) be the Schubert variety of (d - 3)-planes containing xj which is codimension three in Gr(d - 3,P ). Consider |"|*=1a3(x¡) consisting of (d - 3)-planes which contain the plane SK spanned by xx, ... ,xK , then f|*=1 a3(x¡) c fï*=1 a2^i) • ®n tne otrier hand, any A e f|/=i ff2^i) wm ^e m H/Li ai(xi) wnere x,■= A D ti. We therefore have u (Wi>=fW/)- (Xi,...,XK)&XX---XtKi=\ 7=1 We can now consider in 'V f|*=, <72(f,-)as the fiber space over tx x ■■ ■ x tK the product of jc lines with a fiber f|*=, o3(x¡) over a point (x,, ... ,xK) E tx x t2 x ■■ ■ x tK , i.e., each fiber consisting of general member of f)K¡=xa3(x¡). We claim that C\K¡=Xo3(x¡) is a nonempty irreducible subvariety of Gr(d-?>, P ) of dimension 3(d —k —2) whenever d - k - 2 > 0 and 2/c - 2 < d - 1 . To see this, choose (xx, ... , xK) E tx x ■■ ■ x tK . Then we claim that {xx, ... ,xK} spans a (k- l)-plane SK . Suppose not, then {xx, ... ,xK} would span at most a (k - 2)-plane and together with px, ... ,pK at most a (2k - 2)-plane since we may assume that no xi is on C, which meets C at 2k points counting multiplicities. But any «-plane, n < d - 1, meets a rational normal curve of degree d in at most n + 1 points. This gives a contradiction once we have 2k - 2 < d - 1 . Call SK the (k - l)-plane spanned by xx, ... ,x . Considering the projection ns : P —►P from SK we can easily check that D/L] <*T,{Xj)— Gr(d-K-3, P ~K), which confirms that |~|*=ia^i*,) is irreducible of dimension 3(d-K-2). (Note that D^i^^i) = {^K} when 7Í-rC-3 = -l.) Since the base space tx x ■■ ■ x tK and the fibers are irreducible, so is the total space n*=iai^i) ■ Q-E-D- (2.2) Theorem. Assume that k < min(d - 4, j(d - 1)). Then there exists only one component of V(d,5,k) which contains V(d, \(d - l)(d - 2) - k,k) in its closure. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 168 PYUNG-LYUN KANG Proof. Through a point E in V(d, \(d - l)(d - 2) - k,k) , there exist / t(d-l)(d-2)-K \ \{(d- l)(d-2)-K-ö) sheets of the closure of V(d,S,k) each of which corresponds to smoothing some j(d - l)(d - 2) - k - S nodes.
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