Collectanea Mathematica (electronic version): http://www.mat.ub.es/CM
Collect. Math. 52, 2 (2001), 117–126
c 2001 Universitat de Barcelona
Rational projectively Cohen-Macaulay surfaces of
maximum degree
Marina Mancini
Dipartimento di Matematica, Università di Bologna, P.za S. Donato 5
40127 Bologna, Italy
E-mail: mancinm@libero.it
Received May 15, 2000. Revised April 2, 2001
Abstract
In this paper we determine the greatest degree of a rational projectively CohenMacaulay (p.C.M.) surface V in PN and we study the surfaces which attain such
maximum degree.
Introduction
N
Given a nondegenerate smooth surface V embedded in PN
C =P as a projectively CohenMacaulay (p.C.M.) surface, the first problem we study in this paper is to determine a
bound for the degree deg V of V .
If pa denotes the arithmetic genus of the surface V (which, if V is p.C.M., has to
be equal to the geometric genus pg ), we show that
N
N − 1 ≤ deg V ≤
+ pa (V ) − h2 (OV (1)).
2
In particular, when V is rational the greatest degree possible is deg V = N2 .
There are rational p.C.M. surfaces V in PN with such degree, namely the White
Surfaces (see [12]), which are (to our knowledge) the only known ones. Hence it
is quite natural to consider the following problem: are the White Surfaces the only
p.C.M. surfaces of degree N2 in PN ?
In Section 3 we show that a rational surface V , with deg V = N2 and sectional
genus g = N 2−1 , is p.C.M. if its homogeneous ideal contains no quadric.
Keywords: Rational surfaces, embeddings, Cohen-Macaulay
MSC2000: 14JXX.
117
118
Mancini
In particular, we consider the surfaces whose minimal model is P2 and we narrow
the possibilities for other candidates other than White Surfaces as rational p.C.M.
surfaces of maximum degree.
Another problem we investigate is the degree of the generators of the ideal IV of
V . We show that, when V is rational, IV can always generated by forms of degree
≤ 3.
Finally, I thank Prof. A. Gimigliano for his useful suggestions which allowed me
to complete this work.
1. On the degree of p.C.M. surfaces and on their defining ideal
Let V be a nondegenerate smooth surface in PN . We say that V is p.C.M. in PN if its
homogeneous coordinate ring is Cohen-Macaulay.
We recall that in terms of the cohomology of the ideal sheaf of V this fact can be
expressed by saying that h1 (IV (n)) = h2 (IV (n)) = 0, for all n ∈ Z.
If H is a smooth hyperplane section of V of genus g(H), by the Riemann-Roch
Theorem, we have:
h0 (OV (1)) − h1 (OV (1)) + h2 (OV (1)) =
1
(deg V − H.KV ) + 1 + pa (V )
2
(1)
From the adjunction formula we know that 2g(H) − 2 = H 2 + H.KV , hence, by
substituting H.KV = 2g(H) − 2 − deg V in (1), we obtain:
h0 (OV (1)) − h1 (OV (1)) + h2 (OV (1)) = deg V − g(H) + 2 + pa (V ).
(2)
Lemma 1.1
Let V be a nondegenerate smooth surface in PN . If V is p.C.M. then:
a) h0 (OV (1)) = N + 1;
b) h1 (OV (1)) = 0;
c) N = deg V − g(H) + 1 + pa (V ) − h2 (OV (1)).
Proof. a) and b) follow from the exact sequence
0 → IV (1) → OPN (1) → OV (1) → 0.
By substituting h0 (OV (1)) = N + 1 and h1 (OV (1)) = 0 in (2), we obtain c). Lemma 1.2
Let H be a smooth hyperplane section of a nondegenerate p.C.M. surface V ⊆ PN .
Let deg H and g(H) denote, respectively, the degree and the genus of H. Then:
a) h0 (OH (1)) = N ;
b) h1 (OH (1)) = N − 1 − deg H + g(H) = pa (V ) − h2 (OV (1));
c) if pa (V ) = 0, then h2 (OV (1)) = 0 and h1 (OH (1)) = 0.
Rational projectively Cohen-Macaulay surfaces of maximum degree
119
Proof. a) It follows from the exact sequence
0 → IH (1) → OPN −1 (1) → OH (1) → 0.
b) By using the Riemann-Roch Theorem on H we get
h1 (OH (1)) = h0 (OH (1)) − deg H − 1 + g(H) = N − 1 − deg H + g(H).
So, by c) of Lemma 1.1, h1 (OH (1)) = pa (V ) − h2 (OV (1)).
c) Clearly, when pa (V ) = 0, h2 (OV (1)) = 0 and so h1 (OH (1)) = 0. Proposition 1.3
Let V be a nondegenerate p.C.M. surface in PN of arithmetic genus pa . Then for
the degree of V we have the bounds
N
N − 1 ≤ deg V ≤
+ pa (V ) − h2 (OV (1)).
2
Proof. It is well known that, if a nonsingular closed subvariety of PN is p.C.M., then its
hyperplane sections are p.C.M. too. Thus, if H denotes a smooth hyperplane section
of the p.C.M. surface V ⊆ PN , then H is projectively normal in PN −1 .
So, by what we showed in [9], on the genus g(H) of H we have the bound g(H) ≤
N −1
+ 2h1 (OH (1)).
2
Then, from the point c) of Lemma 1.1 and by b) of Lemma 1.2, we deduce that
deg V = N + g(H) − 1 − pa + h2 (OV (1))
N −1
≤
+ N + 2h1 (OH (1)) − 1 − pa + h2 (OV (1))
2
N
=
+ pa − h2 (OV (1)).
2
Since a nondegenerate surface V ⊆ PN has degree deg V ≥ N − 1 (e.g. see [13]),
for the degree deg V of V we have the bounds N − 1 ≤ deg V ≤ N2 + pa − h2 (OV (1)),
as required. Corollary 1.4
If V is a nondegenerate p.C.M. surface in PN of arithmetic genus pa = 0 and
degree deg V , then
N
N − 1 ≤ deg V ≤
.
2
Proof. This follows immediately from Lemma 1.2 and Proposition 1.3. 120
Mancini
Remark. If V is a nondegenerate p.C.M. surface in PN , with arguments similar to
the ones we used above, it is immediate to see that the arithmetic genus pa and the
geometric genus pg of V coincide. In fact, since h2 (IV ) = 0, from the exact sequence
0 → IV → OPN → OV → 0,
we deduce that the irregularity q = pg − pa = h1 (OV ) of V is zero. Hence pa = pg .
Notice that, if V is ruled over a curve C, since q = g(C) (e.g. see [8]), then V is
rational.
Proposition 1.5
The homogeneous ideal of a nondegenerate p.C.M. surface V in PN can be
generated by forms of degree ≤ 3 if pa = h2 (OV (1)), by forms of degree ≤ 4 if
h2 (OV (1)) < pa ≤ N − 1 + h2 (OV (1)).
Proof. We study the problem for a generic hyperplane section H of V .
From our previous work (see [10]) we know that the ideal of the projectively
normal curve H can be generated in degree ≤ 3 if h1 (OH (1)) = 0 and in degree ≤ 4 if
1 ≤ h1 (OH (1)) ≤ N − 1.
Since h1 (OH (1)) = pa (V ) − h2 (OV (1)), by b) of Lemma 1.2, we obtain what we
required. Notice that, by Proposition 1.5, any rational smooth surface embedded in PN as
a p.C.M. surface has its ideal generated by forms of degree ≤ 3.
We recall that this result was already known for certain embedding of blowingups of P2 at a set of distinct points. Namely for all the projective embeddings of the
blowing-ups Xs of P2 at s distinct points via very ample linear systems |tE0 − m1 E1 −
... − ms Es |, with t ≥ σ, where σ is the smallest integer m for which the first difference
∆HZ (m) = 0 of the Hilbert function of Z is zero (see [7; Corollary 3.5]).
2. Embedding of blowing-ups of P2 at a set of distinct points
ms
1
Consider the homogeneous ideal I = pm
... ps ⊂ C[x0 , x1 , x2 ], where p1 , ..., ps
1
are the homogeneous prime ideals corresponding to s distinct points, P1 , ..., Ps , of P2 .
Let Z = (P1 , ..., Ps ; m1 , ..., ms ) be the subscheme of P2 associated to I (usually
called a scheme of fat points), where we assume m1 ≥ ... ≥ ms .
On the blowing-up Xs of P2 at the Pi s, i = 1, ..., s, we denote by |Dt | = |tE0 −
m1 E1 − ... − ms Es | the complete linear system corresponding to the linear system
St (P1 , ..., Ps ; m1 , ..., ms ) of all the plane curves of degree t passing through each Pi
with multiplicity at least mi , i = 1, ..., s (see [8]).
Since every mi -fold point impose mi2+1 conditions on curves of St (P1 , ..., Ps ;
m1 , ..., ms ), if the dimension of the linear system is equal to
s mi + 1
t+2
− 1, 0 ,
max
−
2
2
i=1
then we shall say that our system is regular.
Rational projectively Cohen-Macaulay surfaces of maximum degree
121
We recall that, in terms of the cohomology of the ideal sheaf IZ of Z, the regularity
of St (P1 , ..., Ps ; m1 , ..., ms ) is expressed by the condition h0 (IZ (t)) · h1 (IZ (t)) = 0 (e.g.
see [4]).
Remark. If the complete linear system |Dt | defines an embedding of Xs in PN as a
p.C.M. surface V , then the linear system St (P1 , ..., Ps ; m1 , ..., ms ) is regular.
In fact, since V is p.C.M. in PN , h2 (IV (1)) = 0, and from the exact sequence
0 → IV (1) → OPN (1) → OV (1) → 0,
we get that h1 (OV (1)) = h1 (OXs (Dt )) = 0.
But h1 (OXs (Dt )) = h1 (IZ (t)) (e.g. see [5]), where Z is given as above, and so we
are done.
Now, let HZ (t) be the Hilbert function of the scheme Z and let σ = σ(Z) be the
smallest integer t for which ∆HZ (t) = HZ (t) − HZ (t − 1) = 0.
If τ = τ (Z) denotes that smallest t for which h0 (IZ (t)) · h1 (IZ (t)) = 0, i.e.
such that the linear system St (P1 , ..., Ps ; m1 , ..., ms ) is regular, it is well known that
σ = τ + 1. So, by what we have seen above, in order to have an embedding of Xs in
PN via |Dt | as a p.C.M. surface, it must be t ≥ τ = σ − 1.
Now let us consider the following result:
Proposition 2.1 (See [7]).
Suppose that no line of P2 has intersection of degree ≥ σ = τ +1 with Z. Then |Dt |
embeds the blowing-up Xs of P2 at the points of Z as a p.C.M. surface if t ≥ σ = τ + 1.
We could replace “if” with “if, and only if” in the statement of the proposition
above if we can prove that |Dσ−1 | = |Dτ | doesn’t embed Xs in PN as a p.C.M. surface.
We are able to do that when the scheme Z is a set of generic points of P2 .
Proposition 2.2
Let Z = {P1 , ..., Ps } be a set of generic points in P2 and let s = t+1
2 +k, 0 < k < t
and t > 3. Then the linear system |Dt | = |tE0 − E1 − ... − Es | defines an embedding
of Xs as a p.C.M. surface if, and only if, t ≥ σ.
Proof. We have only to prove that |Dσ−1 | = |Dτ | doesn’t determine an embedding of
Xs as a p.C.M. surface.
It’s easy to see that the smallest integer m for which the first difference ∆HZ (m)
of the Hilbert function of Z is zero is σ = τ + 1, hence τ = t.
Now, let us suppose that the complete linear system |Dτ | = |Dt | = |tE0 − E1 −
... − Es | determines an embedding in Pt−k of the blowing-up Xs of P2 at the Pi s,
i = 1, ..., s.
In order to show that the embedded surface V ⊆ Pt−k is not p.C.M. it’s enough
t−1to
prove that a generic hyperplane
section
H
of
V
,
which
is
a
curve
of
genus
g(H)
=
2
and degree deg H = t2 − s = 2t − k, is not a projectively normal curve in Pt−k−1 .
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Mancini
From the exact sequence
0 → IH (2) → OPt−k−1 (2) → OH (2) → 0,
we get:
h0 (IH (2)) = h0 (OPt−k−1 (2)) − h0 (OH (2)) + h1 (IH (2))
t−k+1
=
− 2deg H + 1 − g(H) + h1 OH (2) + h1 (IH (2)).
2
Since 2deg H > 2g(H) − 2 for all k < t, we have h1 (OH (2)) = 0 and a simple
computation gives
0
h (IH (2)) =
k+2
− tk − 1 + h1 (IH (2)).
2
But the term k+2
− tk − 1 is negative, for all k ∈ Z such that 0 < k < 2t − 3.
2
Hence, with 0 < k < t and t > 3, by hypothesis, h1 (IH (2)) cannot be zero. Thus H is
not projectively normal in Pt−k−1 , as required.
Notice that, for k = 0, τ = t − 1 and, since no curve of degree t − 1 contains
P1 , ..., Ps , we have |Dτ | = ∅. When Z is a scheme of fat points we don’t know, in general, what is τ = σ − 1,
hence we cannot check, as in Proposition 2.2, if |Dτ | = |Dσ−1 | defines a p.C.M. surface
in PN .
3. Rational p.C.M. surfaces of maximum degree
It is well known (see [13]) that if V is a surface embedded in PN with degree deg V ,
then deg V ≥ N − 1 and that the smooth surfaces of degree deg V = N − 1 are either
rational normal scrolls or the Veronese Surface in P5 .
Now, by what we proved in the previous sections, we also know the greatest degree
of an embedding of a rational smooth surface V in PN as a p.C.M. surface, namely
deg V = N2 .
In this section we want to investigate the rational p.C.M. surfaces in PN of maxi mum degree N2 .
First of all we want to determine as many properties as possible of these surfaces
whose sectional genus, by Lemma 1.1, is g(H) = N 2−1 .
Proposition 3.1
The homogeneous ideal IV of a rational p.C.M. surface V of degree
generated in degree exactly 3.
N 2
in PN is
Rational projectively Cohen-Macaulay surfaces of maximum degree
123
Proof. By Proposition 1.5 we know that the ideal IV of V can be generated by forms
of degree ≤ 3, hence, since h1 (IV (2)) = 0, it is enough
to show that it doesn’t contain
N +2
0
any quadratic form, i.e. that h (OV (2)) = 2 .
Consider the exact sequence
0 → OV (1) → OV (2) → OH (2) → 0,
where H denotes a smooth hyperplane section of V .
0
1
= N +1 and h
(OV (1)) = 0, thus h0 (OV (2)) =
Lemma 1.1, we0 have h (OV(1))
N +2By
N +2
if and only if h (OH (2))
= N2+1 .
2
N1−1
N = 2 − N −
= 2 2 , we have h1 (OH (2)) = 0 and so
(2))
= 2 2 > 2g(H)
Since deg(O
H
h0 (OH (2)) = 2 N2 − N 2−1 + 1 = N2+1 , as required. Proposition 3.2
Let V ⊆ PN be a rational smooth surface of degree deg V = N2 and sectional
genus g(H) = N 2−1 . If h1 (OV (1)) = 0, then V is p.C.M. if, and only if, it is projectively normal.
Proof. We want to show that h1 (OV (m)) = 0, for all m ≥ 0, from which we immediately have h2 (IV (m)) = 0, for all m ≥ 0.
Consider the exact sequence
0 → OV (m − 1) → OV (m) → OH (m) → 0.
(3)
Clearly h1 (OV ) = 0, while h1 (OV (1)) = 0, by hypothesis.
Simple computations prove that mdeg H = m N2 > 2 N 2−1 − 2 = 2g(H) − 2, for
all m ≥ 2, thus h1 (OH (m)) = 0, for all m ≥ 2.
With h1 (OV (1)) = 0, from (3) we have that h1 (OV (m − 1)) = 0 implies
h1 (OV (m)) = 0, for all m ≥ 2, as required. There are known rational p.C.M. surfaces which attain the maximum degree,
namely the White Surfaces,
which are defined by the embedding in Pt of the blowing
t+1
up Xs of P2 at s = 2 generic points P1 , ..., Ps via the complete linear system
|Dt | = |tE0 − E1 − ... − Es | (see [6]).
Since the White Surfaces are the only known examples of rational p.C.M. surfaces
of maximum degree, it is quite natural to put the following.
Question: Are the White Surfaces the only rational p.C.M. surfaces of degree
PN ?
N 2
in
We want to check if there exist other rational surfaces which can be candidates as
p.C.M. surfaces of maximum degree.
In particular we investigate the rational surfaces V ⊆ PN whose minimal model
is P2 .
First of all notice that V cannot be minimal, since the linear system of allthe plane
curves of degree a ≥ 2 on P2 determines an embedding
of
P2 in PN , N = a+2
− 1,
2
with degree a2 which is certainly strictly smaller that N2 .
124
Mancini
Now, let Xs denote the smooth surface obtained from P2 by s consecutive blowingups, each of them at one point.
Then we can consider the embedding of Xs via a very ample regular linear system
|Dt | = |tE0 − E|, where E = m1 E1 + ... + ms Es .
Let τ (E) = min t/h1 (OXs (Dt ) = 0 .
Thus we have:
Proposition 3.3
Let |Dt | = |tE0 − m1 E1 − ... − ms Es | be a very ample regular linear system on
Xs . If |Dt | defines a p.C.M. surface V ⊆ PN of maximum degree N2 , then either Dt
is as in the White Surfaces case (hence t = τ (E) + 1) or t = τ (E).
Proof. Let us suppose that t ≥ τ (E) + 1.
Then the vector space H 0 (OXs (Dt−1 )), by the Riemann-Roch Theorem on Xs ,
s mi +1
has dimension t+1
− i=1 2
≥ 0.
2
s
2
2
Since deg V = t − i=1 mi = N2 and 2g(Dt ) − 2 = Dt .(Dt + KXs ), hence
N −1
N
N 2 − 5N
mi = 2
−
− 2 + 3t =
+ 3t,
2
2
2
i=1
s
we have
t2 + t
−
dim H OXs (Dt−1 ) =
2
0
s
i=1
mi 2 +
2
s
i=1
mi
= N − t ≥ 0, i.e. t ≤ N.
s
s
When mi = 1 for some 1 ≤ i ≤ s, i=1 mi 2 > i=1 mi , hence t2 − 3t − N 2 +
3N > 0.
Solving the quadratic equation we find t > N and so we get a contradiction. Hence
if some mi = 1, we must have t = τ (E).
If mi = 1, for all 1 ≤ i ≤ s, either we are in the White Surfaces case, or in the
making of Xs we are blowing-up some point on an exceptional divisor; in this last case
it is well known that Dt cannot be very ample, hence this case cannot happen. We can check that what is really needed to have that V ⊆ PN is p.C.M. is that it
is not contained in quadrics.
Proposition 3.4
Let V ⊆ PN be a nondegenerate smooth surface of degree N2 and sectional genus
N −1
. Suppose that h0 (OV (1)) = N + 1 and that h1 (OV (1)) = 0. If h0 (IV (2)) = 0,
2
then V is p.C.M.
Rational projectively Cohen-Macaulay surfaces of maximum degree
125
Proof. Let us suppose that h0 (IV (2)) = 0 (otherwise we have nothing to prove).
Since
h0 (OV (1)) = N + 1
and h1 (OV (1)) = 0, we have
h1 (IV (1)) = h0 (IV (1)) − h0 (OPN (1)) + h0 (OV (1)) = 0
and
h (IV (2)) = h (OV (2)) − h (OPN (2))
1
0
0
= h0 (OV (1)) + h0 (OH (2)) − h0 (OPN (2)) = 0.
We want to show that having h1 (IV (1)) = 0 = h1 (IV (2)) is enough to deduce
that the surface V is p.C.M. in PN .
Let H denote a smooth hyperplane section of the surface V . From the exact
sequence
0 → OV (m − 1) → OV (m) → OH (m) → 0
we know that h1 (OH (1)) = 0, hence we can apply the following lemma.
Lemma 3.5 (E.g. see [1]).
If C ⊆ Pr is a smooth curve for which OC (1) is non special, then C is projectively
normal if, and only if, it is linearly and quadratically normal.
Consider the exact sequence
0 → IV (m) → IH (m) → IH,V (m) → 0.
When m = 1, 2, we find, respectively, h1 (IH (1)) = 0 and h1 (IH (2)) = 0. Hence
H is projectively normal in PN −1 .
This implies that V is projectively normal in PN (see [3]) and so, by Proposition 3.2, V is p.C.M., as required. Simple computations prove that, when
t = τ (Z), we could
find
linear systems
which define p.C.M. surfaces of degree N2 and sectional genus N 2−1 in PN . A possible
candidate is:
|DN +1 | = (N + 1)E0 − 2E1 − ... − 2EN −1 − EN − ... − E(N )+4 .
2
This linear system, in fact, is very ample for all N ≥ 5, by [2; Theorem 1.1], and
regular (e.g. see [4]).
Considering Proposition 3.4, the first meaningful problem is the following.
Problem: Does the homogeneous ideals IV of the surfaces V ⊆ PN defined by |DN +1 |
above contain quadrics?
Remark. In P5 we can determine (see [11]) all the rational smooth surfaces of degree
10 and sectional genus 6, which are candidates as p.C.M. surfaces (it is still an open
problem to check if all of them actually exist).
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Mancini
The pairs (Xs , Dt ) and (Fse , D), where Fse denotes the blowing-up at s distinct
points of a Hirzebruch surface Fe , corresponding to these surfaces are:
(i) (X15 , 5E0 − E1 − ... − E15 ) (White Surface);
(ii) (X14 , 6E0 − 2E1 − ... − 2E4 − E5 − ... − E14 );
(iii) (X12 , 7E0 − 2E1 − ... − 2E9 − E10 − ... − E12 );
(iv) (X10 , 10E0 − 3E1 − ... − 3E10 );
(v) (X15 , 6E0 − 3E1 − 2E2 − E3 − ... − E15 ) (equiv. F14
0 , 3C0 + 4f − E1 − ... − E14 );
10
(vi) (X11 , 8E0 − 3E1 − 3E2 − 2E3 − ... − 2E11 ) (equiv. F0 , 5C0 + 5f − 2E1 − ... − 2E10 );
(vii) (X12 , 9E0 − 3E1 − ... − 3E7 − 2E8 − E9 − ... − E12 );
(viii) (X11 , 9E0 − 3E1 − ... − 3E6 − 2E7 − ... − 2E10 − E11 );
(ix) (F14
2 , 3C0 + 7f − E1 − ... − E14 );
(x) (F12
e , 4C0 + (2e + 5)f − 2E1 − ... − 2E6 − E7 − ... − E12 ; 0 ≤ e ≤ 2);
(xi) (F11
e , 4C0 + (2e + 6)f − 2E1 − ... − 2E9 − E10 − E11 ; 0 ≤ e ≤ 2).
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