AbstractWe prove the existence of a coarse, irreducible moduli space Cm−1g (resp. Cm−1g,1) for (resp. pointed) subcanonical extremal curves of level m−1 and genus g in PCr. When g=3r, r≥4, r≠5, odd (resp. even) we show that C23r (resp. C23r,1) is rational

Casnati, Gianfranco

Del Centina, A.

CASNATI, GIANFRANCO

DEL CENTINA A.

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

The rationality of certain moduli spaces associated to half-canonical extremal curves

Casnati, G.

Centina, A.Del

Elsevier - Publisher Connector 

The rationality of certain moduli spaces associated to half-canonical extremal curves 

PORTO Publications Open Repository TOrino

Journal of Pure and Applied Algebra 141 (1999) 201{209
www.elsevier.com/locate/jpaa
The rationality of certain moduli spaces associated
to half-canonical extremal curves
G. Casnati a;, A. Del Centina b;1
aDipartimento di Matematica Pura ed Applicata, Universita Degli Studi di Padova, via Belzoni 7,
35131 Padova, Italy
bDipartimento di Matematica, Universita Degli Studi di Ferrara, via Machiavelli 35,
44100 Ferrara, Italy
Communicated by A.V. Geramita; received 14 July 1997; received in revised form 28 December 1997
Abstract
We prove the existence of a coarse, irreducible moduli space Cm−1g (resp. C
m−1
g; 1 ) for (resp.
pointed) subcanonical extremal curves of level m− 1 and genus g in PrC . When g=3r, r 4,
r 6=5, odd (resp. even) we show that C23r (resp. C23r; 1) is rational. c© 1999 Elsevier Science B.V.
All rights reserved.
MSC: 14H10; 14H45
0. Introduction
A classical result states that the canonical model of a non-hyperelliptic trigonal curve
C of genus g 4 lies on a unique rational normal scroll X PrC and C \Sing(X )= ;
(see [1, Ch. III]).
If C is general and its genus g 6 is even then X =P1CP1C , and C 2 j3f1 + ((g+
2)=2)f2j, where f1; f2 are two non-skew lines on X . Thus, with the above hypotheses,
the trigonal locus TgMg inside the moduli space of smooth curves of genus g, is
birationally equivalent to the quotient
P(H 0(X;OX (3f1 + ((g+ 2)=2)f2)))=(SL2SL2):
 Corresponding author. Supported by the framework of the SCIENCE contract no. SCI-0398-C(A).
E-mail address: casnati@galileo.math.unipd.it (G. Casnati)
1 Partially supported by MURST 40% funds.
0022-4049/99/$ - see front matter c© 1999 Elsevier Science B.V. All rights reserved.
PII: S 0022 -4049(98)00037 -1
202 G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209
N.I. Shepherd-Barron in his paper [10] shows that if g 2 (mod 4) and g 6 the
above quotient is rational.
The aim of this paper is to generalize the methods used in [10] to a class of curves
whose geometry is quite similar to that of trigonal curves, namely the class of sub-
canonical extremal curves.
Let C PrC be a smooth, irreducible, connected curve of genus g and degree d.
Then the following inequality, called Castelnuovo’s bound, holds for C:
g (d; r) :=

m
2

(r − 1) + m (0.1)
where d=m(r − 1) +  + 1 and 0  r − 2 (see [1, p. 116]). If equality holds in
formula (0.1) then C is called extremal curve.
Notice that if C is extremal and d=2r then formula (0.1) gives r= g − 1 (m=2,
=1) and therefore C is a canonical curve since extremal curves are projectively
normal (see [1, p. 117]).
Assume from now on that r 3 and d 2r + 1. Then extremal curves C PrC of
degree d do exist and, if r 6=5 and >0, each such curve is a smooth element of the
linear system
j(m+ 1)H − (r − − 2)fj (0.2)
on a rational normal scroll X PrC (H and f denote the hyperplane section and the
general bre of X : see [1, Ch. III, Theorem 2.5(iii)]. Moreover, we remark that the
bres of X cut out on C a g1m+1.
With the above conditions on r and  it follows from Theorem 1.4(i) of [2] that
the Zariski open subset H0d; g; r of the Hilbert scheme of curves of degree d and genus
g= (d; r) in PrC , and whose points correspond to extermal curves of type 0.2, is
irreducible and one has
dimH0d; g; r =(r − 1)

m+ 1
2

+ r + 2

+(+ 2)(m+ 2)− 4: (0.3)
If =1 we also have that jKC j= jOC(m − 1)j (see [1, Ch. III, Corollary 2.6(iii)]).
According to Section 2 of [3] such curves will be called subcanonical extremal curves
of level m− 1.
The group PGLr+1 acts on H0d; g; r , thus we get a PGLr+1-equivariant rational map
h :H0d; g; r !Mg into the moduli space of smooth curves of genus g= (r; d). Since
it follows from Proposition 2.1 of [3] that the grd inducing the embedding C PrC is
unique and complete, then the bre over the point [C]2Mg representing C is exactly
the PGLr+1-orbit of [C]2H0d; g; r .
In particular, we can dene in a natural way a coarse moduli space for subcanonical
extremal curves of level m− 1 and genus g, namely Cm−1g := im(h)Mg.
The above remarks show that H0d; g; r=PGLr+1Cm−1g (we will denote by = isomor-
phisms and by  birational equivalences). Since the action of PGLr+1 on H0d; g; r has
nite stabilizer, formula (0.3) and the above discussion yields the following.
G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209 203
Proposition 0.4. Let r 3; r 6=5. There exists a coarse; irreducible moduli space
C
m−1
g of dimension
dimCm−1g =(r − 1)

m+ 1
2

+ r + 2

+3m+ 3− (r + 1)2
for subcanonical extremal curves of level m− 1 and genus g in PrC .
Let Mg;1 be the moduli space of smooth pointed curves of genus g, let m :Mg;1!Mg
be the natural projection and set Cm−1g;1 :=m
−1(Cm−1g ). Since the bres of m are irre-
ducible the same is true for Cm−1g;1 .
Corollary 0.5. Let r 3; r 6=5. Cm−1g;1 is a coarse irreducible moduli space for smooth
pointed subcanonical extremal curves of level m − 1 and genus g in PrC . Moreover
dimCm−1g;1 = dimC
m−1
g + 1.
The aim of this paper is to prove the following theorem.
Theorem 0.6. If r 4; r 6=5 is odd (resp. even) then C23r (resp. C23r;1) is rational of
dimension 5r + 3 (resp. 5r + 4).
We remark that for r=2 half-canonical extremal curves are smooth plane quintic.
The rationality of their coarse moduli space has been proved by N.I. Shepherd-Barron
in [11].
1. A birational model of C23r
In this section we describe a birational model of C23r . Let C be as in the introduction.
We already noticed that the curve C carries a gr3r−1 which is unique and complete.
Moreover, C lies on a unique rational normal scroll X (of minimal degree r − 1 in
PrC) and C does not intersect its singular locus Sing(X ).
Blowing up X along Sing(X ) we obtain a desingularization map h :=P(OP1C OP1C
(−h)) ’−!X for some h 0. Let h :h!P1C be the canonical projection and denote
by f anyone of its bres (notice that f=P1C and f2 = 0). If h>0 (resp. h=0) there
exists a unique irreducible, smooth, rational curve (resp. pencil of curves) D0h
(resp. jD0j Div(h)) such that D20 =−h and D0f=1. Moreover, the map h
’−!X
is induced by the linear system jD0 +‘fj for some ‘ h and X is singular if and only
if ‘= h. Since r − 1=deg(X )= (D0 + ‘f)2 then we also have
2‘ + 1= h+ r; h r + 1 (mod 2): (1.1)
The curve C is embedded in h as a quadrisecant curve, more precisely C 2 j4D0 +
(r+2h+1)fj (see [4, Theorem 5.4 and its proof]). Conversely each smooth, connected
204 G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209
curve C 2 j4D0 + (r + 2h+ 1)fj is embedded in PrC via ’jC as an extremal curve of
degree 3r − 1. In particular, we can dene rational maps
’h : j4D0 + (r + 2h+ 1)fj!C23r :
Proposition 1.2. Let h=0; 1 and r 3; r 6=5. Then im(’h) is dense in C23r for r h+1
(mod 2).
Proof. It suces to check that dim(im(’h))= 5r + 3, since C
2
3r is irreducible of
dimension 5r + 3. To this purpose let us consider the group Aut(h). In any case
dim(Aut(h))= 6 (see [12]).
As in the introduction notice that the embedding C h is induced by the unique
gr3r−1. In particular, each automorphism of h leaving C xed, carries such a g
r
3r−1
into itself. It follows that the bre of ’h over a point [C]2 im(’h) are the G-orbits
of the divisor C with respect to G :=Aut(1) if h=1, G :=Aut(0)=O4 if h=0
and r=3 or G := SO4Aut(0)=O4 if h=0 and r 7. In any case dim(G)= 6 (see
again [12]), hence
dim(im(’h)) dim j4D0 + (r + 2h+ 1)fj − 6=5r + 3
since h0(h;Oh(C))= h
0(P1C; hOh(C)) (see [7, Lemma V 2.4]).
Now assume that r 3 is odd. Then it follows from formulas (1.1) that h is even
and ‘ 1 thus X =P1CC P1C is smooth. Moreover Proposition 1.1 and its proof yield
the birational equivalence
C
2
3r  j4D0 + (r + 1)fj=G;
where G=O4 if r=3 and G= SO4 if r 7. Notice that j4D0+(r+1)fj =P(V4⊗V 0r+1)
where Vd (resp. V 0d) is the set of forms of degree d in the variables x1 and x2 (resp.
y1 and y2). On the other hand, the representation of SO4 on V4⊗V 0r+1 is equivalent
to the natural representation of SL2SL2 on V4⊗V 0r+1. We summarize results in the
following:
Theorem 1.3. If r=2k − 1 7 then there exists a birational equivalence
C
2
3r P(V4⊗V 02k)=(PSL2PSL2):
2. The rationality of P(V4⊗V 02k)=(PSL2PSL2)
In this section, following the proof of Proposition 6 of [10], we will prove
Theorem 2.1. P(V4⊗V 02k)=(PSL2PSL2) is rational if k  3.
A immediate consequence of Theorem 2.1 above is the rationality of C23r for odd
r 7, stated in Theorem 0.4 of the introduction.
G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209 205
Let f2V4⊗V 02k . Using the symbolical notation (see [6]) we can write f= a4x ⊗A2ky =
b4x ⊗B2ky where ax := a1 x1 + a2 x2, bx := b1 x1 + b2 x2, Ay :=A1y1 + A2y2, By :=B1y1 +
B2y2. We dene a covariant
’ :V4⊗V 02k !V4⊗V 00 ;
f! (ab)2(AB)2ka2x b2x ;
where, as usual, (ab) := a1b2−a2b1, (AB) :=A1B2−A2B1. ’ induces a rational (PSL2
PSL2)-equivariant map
 :P(V4⊗V 02k)!P(V4⊗V 00):
Let  : eP!P(V4⊗V 02k) be the blow up of P(V4⊗V 02k) along the base locus of  and
let e : eP!P(V4⊗V 00)
be the induced morphism. Notice that e is again a (PSL2PSL2)-equivariant map.
The following proposition is an easy consequence of the denition of  and its proof
is a trivial and tedious computation.
Proposition 2.2. If f=
P4
i=0
P2k
j=0(−1) j
(2k
j

i; j x4−i1 x
1
2 ⊗y2k−j1 yj2 then
 (f)=
4X
h=0
h x4−h1 x
h
2 ;
where
0 = 2
2kX
j=0
(−1) j

2k
j

(0; j2;2k−j − 1; j1;2k−j);
1 = 4
2kX
j=0
(−1) j

2k
j

(0; j3;2k−j − 1; j2;2k−j);
2 = 2
2kX
j=0
(−1) j

2k
j

(0; j4;2k−j + 21; j3;2k−j − 32; j2;2k−j);
3 = 2
2kX
j=0
(−1) j

2k
j

(4; j1;2k−j − 2; j3;2k−j);
4 = 2
2kX
j=0
(−1) j

2k
j

(4; j2;2k−j − 3; j3;2k−j):
206 G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209
Corollary 2.3. Both  and e are dominant.
Proof. It suces to prove that  is dominant. Let
F := ff2P(V4⊗V 02k) j4; k = 0; k = ; 2; k = ; i; j =0
for (i; j) 6=(4; k); (0; k); (2; k)g:
It follows from Proposition 2.2 that  (f)=
P4
h=0 h x
4−h
1 x
h
2 where 1 = 3 = 0 and
0 = 4 = 2(−1)k

2k
k

; 2 = 2(−1)k

2k
k

(2 − 32):
In particular,  (F) is (PSL2PSL2)-dense in P(V4⊗V 00) (see [9, Section 1]), whence
for each general f2P(V4⊗V 00), there exists g2PSL2PSL2 such that g(f)2  (F),
thus there exists f2F such that  (f)= g(f) hence f=  (g−1(f)).
Let
U :=P(hx41 − x42 ; x31 x2; x1 x32 i⊗V 00);
V :=P(hx41 + x42 ; x21 x22 i⊗V 00);
dene
e :=
 
1 0
0 1
!
; a :=
 
i 0
0 −i
!
; b :=
 
0 1
−1 0
!
;
and let H PSL2 be the subgroup generated by the classes of e, a, b. The nor-
malizer N of H inside PSL2 is the stabilizer of V (see [9, Section 1]), thus V is
a (PSL2PSL2; NPSL2)-section of P(V4⊗V 00) in the sense of [8].
Now let Y := (e −1(V )).
Proposition 2.4. If k  2 then Y is irreducible.
Proof. Let L :=P(V4⊗V 02k) the subspace dened by the equations
1; i= 3; j = 0; l − 4; l=0; i; j; l=0; : : : ; 2k:
The equations of Y P(V4⊗V 02k) are
#1 :=
2kX
j=0
(−1) j

2k
j

(4; j1;2k−j − 2; j3;2k−j);
#2 :=
2kX
j=0
(−1) j

2k
j

(0; j3;2k−j − 1; j2;2k−j);
#3 =
2kX
j=0
(−1) j

2k
j

((0; j − 4; j)2;2k−j − 1; j1;2k−j + 3; j3;2k−j)
(see Proposition 2.2). In particular LY .
G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209 207
Let Y0Y be an irreducible component containing L and let Y1 be another one not
contained in Y0. Obviously Y0 \Y1Sing(Y ), hence L\Y1Sing(Y ).
Since Y is the intersection of three hypersurfaces one also has codim(Y1) 3, thus
codim(L\Y1) 3 + codim(L) 6k + 6: (2.1)
On the other hand, the Jacobian matrix of the #i’s is
J :=
0BBBB@
0 3; i 2; i
4; j −2; j −21; j
−3; h −1; h 0; h − 4; h
−2; l 0; l 23; l
1; m 0 −2; m
1CCCCA
i; j; h; l;m= 0 ;:::;2k
:
In particular, rk(J ) 2 at each point of L\Y1, thus L\Y1X where X P(V4⊗V 02k)
is dened by the equations8>>>><>>>>:
1; i=0;
3; j =0;
0; l − 4; l=0;
(0; h4; h − 22; h)2; h=0;
where i; j; l; h=0; : : : ; 2k. With some easy computations one can check that the above
system is equivalent to the system8>>>><>>>>:
1; i=0;
3; j =0;
0; l − 4; l=0;
((0; h + 4; h)2 − 422; h)2; h=0:
(2.2)
Each polynomial in (2.2) involves a dierent set of variables hence they form a regular
sequence. It follows that codim(X )= 4(2k + 1) so that
codim(L\Y1) 8k + 4: (2.3)
Confronting (2.1) and (2.3) one should have 6k + 6 8k + 4 which is possible only
if k =0; 1.
We conclude that Y =Y0.
Since Y is irreducible then it is a (PSL2PSL2; NPSL2)-section of P(V4⊗V 02k)
(see [8]). Thus we get the following result.
Corollary 2.5. P(V4⊗V 02k)=(PSL2PSL2)Y=(NPSL2).
Let M P(V4⊗V 02k) be the subspace dened by the equations
0; i + 4; i= 2; j =0;
208 G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209
where i; j=0; : : : ; 2k. We denote by L :Y !M the projection from L. If we choose
1; j, 3; h, 0; l − 4; l (i.e. if we x m2M) the equations #i specialized at these values
becomes linear in the remaining variables.
Such equations are generically of maximal rank thus Y is generically a vector-bundle
over M . Moreover L is (NPSL2)-equivariant.
Lemma 2.6. M =P(U ) and the action of NPSL2 on M is almost free for k  3.
Proof. Let p2P(V 02k) such that the condition g(p)=p implies g= id2PSL2 (such
a p exists since 2k  6).
Let f2P(hx41 − x42 ; x31 x2; x1 x32 i) such that the condition g(f)=f implies g= id2N
(such an f exists since N =S4 and the three-dimensional irreducible representations
of S4 are exact: see Section 1 of [9]).
It follows that the stabilizer of f⊗p is trivial.
Let L :Y=(NPSL2)!M=(NPSL2) be induced by L. The method of irreducible
representation (see [5]) implies that L is again a vector bundle.
Let
W := (hx41 − x42 ; x31x2; x1 x32 iC)⊗V 00 V0⊗V 06 :
W is an almost free linear representation of NPSL2. Moreover
P(W )=(NPSL2)P(hx41 − x42 ; x31x2; x1x32 iC)=NP(V 06)=PSL2P1C
which is rational (see [9]: for the rst factor see the corollary in section 1, for the
second one see Theorem 0.1 or 0.2).
Since MP(W )=(NPSL2) is a vector bundle over P(W )=(NPSL2) it is also
rational. On the other hand, MP(W )=(NPSL2) is also a vector bundle over M=(N
PSL2) with typical bre P(W )C10 hence M=(NPSL2)C10 is rational too.
Since −1L (m)=C4k−1 it follows that Y=(NPSL2) is rational as soon as 4k−1 10
i.e. if k  3.
3. Pointed curves
In this section we complete the proof of Theorem 0.6.
Proposition 3.1. If r 4 is even then C23r;1 is rational.
Proof. Let r be even and let C be a subcanonical extremal curve of level 2 and genus
3r contained in the ruled surface 1 as an element of the linear system jD0 +(r+3)fj
(for the notations see Section 1). Contracting D0 to a point, we map C to a plane
curve C0P2C of degree r + 3 with a point of multiplicity r − 1.
G. Casnati, A. Del Centina / Journal of Pure and Applied Algebra 141 (1999) 201{209 209
Fix P;Q2P2C and let W be the linear system of curves in P2C through Q and having
P as a (r − 1)-fold point. We have a rational map
’ :W !C23r;1
sending each curve to its isomorphism class.
Let SL3; P and SL3; Q be the stabilizers inside SL3 of P and Q respectively. The group
G := SL3; P \ SL3; Q  SL3 acts on W (with nite stabilizer). Since ’ is by construction
G-equivariant then
dim’(W )= dimW − dimG=5r + 4;
thus ’ is dominant (see Corollary 0.5).
On the other hand, the orbits of G coincide with the bres of ’ thus C23r;1W=G
which is rational since G is triangular (see [13]).
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The rationality of certain moduli spaces associated to half-canonical extremal curves

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