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Set-theoretic complete intersection monomial curves in P
n
Abstract. In this paper, we give a sufficient numerical criterion for a mono-
mial curve in a projective space to be a set-theoretic complete intersec-
tion. Our main result generalizes a similar statement proven by Keum
for monomial curves in three-dimensional projective space. We also prove
that there are infinitely many set-theoretic complete intersection mono-
mial curves in the projective
n−space for any suitably chosen
n − 1 inte-
gers. In particular, for any positive integers
p, q, where gcd(
p, q) = 1
, the
monomial curve defined by
p, q, r is a set-theoretic complete intersection
for every
r ≥ pq(
q − 1)
.
Mathematics Subject Classification (2010). Primary 14M10;
Secondary 14H45.
Keywords. Set-theoretic complete intersections, Monomial curves.
1. Introduction. Let
m1
< · · · < mn be some positive integers with
gcd(
m1
, . . . , mn) = 1
. A monomial curve
Cm
P
n over an algebraically closed field
K is a curve with generic zero
(
umn, umn−m1
vm1
, . . . , umn−mn−1
vmn−1
, vmn)
where
u, v ∈ K and (
u, v) = (0
, 0)
. The monomial curve
Cm
a
set-theoretic complete intersection (s.t.c.i.) on
f1
, . . . , fn−1 if it is the inter-section of (
n − 1) hypersurfaces defined by nonzero homogeneous polynomials
f1
, . . . , fn−1 in a polynomial ring over
K; that is, if we can write
Cm
Z(
f1
, . . . , fn−1)
. It is known that every monomial curve in P
n is an s.t.c.i.,when the field
K is of characteristic
p > 0
, see The extension to the char-acteristic zero case is a longstanding open problem, besides some special cases.
Robbiano and Valla show that rational normal curves in P
n and arithmeticallyCohen–Macaulay monomial curves in P3 are set-theoretic complete intersec-tions in any characteristic without giving the equations of the surfaces involved
The author is supported by Viet Nam NAFOSTED (National Foundation for Science &
explicitly, see , Keum proves in ] that the monomial curves
Cp,q,r areset-theoretic complete intersections by giving explicitly the polynomials defin-ing the corresponding surfaces, in the special cases where
p = 1 or
q =
r − 1under further mild arithmetic conditions on
r. Moreover, S¸ahin also providesin the equations defining s.t.c.i. symmetric monomial curves in P3 whichare arithmetically Cohen–Macaulay. Even though there are methods producings.t.c.i. monomial curves in P
n starting with an s.t.c.i. monomial curve in P
n−1
,see e.g. ], it is rather difficult to prove that certain families of monomialcurves are s.t.c.i. in P
n and to present the polynomials explicitly defining thehypersurfaces cutting out these curves.
The purpose of this paper is to give a sufficient criterion for mono-
in P
n to be s.t.c.i.’s depending on the arithmetics of
m1
, . . . , mn. Our main result generalizes the main result of and providesthe equations of the hypersurfaces cutting out the curves.
2. The Main Result. In this section we prove our main assertion and list some
of its consequences.
Theorem 2.1. Let m1
< · · · < mn be some positive integers with the prop-
erty gcd(
m1
, . . . , mn) = 1
and satisfying the following two conditions for some
nonnegative integers ai,j:
(I)
mi =
ai,i−1
mi−1
−
m1 = 1 and
ai,i−1
>
m1
> 1 and
ai,i−1
≥
j=1
ai,j mj, for all 3
≤ i ≤ n. Then the monomial curve
Cm1
,.,mn
is a set-theoretic complete intersection on
F1
, . . . , Fn−1
, where
F1 =
xm2
, and for 3
≤ i ≤ n, Fi−1 is given by
(
−1)
mi−1
−k mi−1
x
Proof. First we demonstrate that all the monomials of
Fi, for 3
≤ i ≤ n, havenonnegative exponents. By (I),
mi − kai,i−1 = (
mi−1
− k)
ai,i−1
−
Since
mi−1
− k ≥ 1
, it follows that
mi − kai,i−1
≥ ai,i−1
−
by (II). As for the exponent of
x0
, we have
ai,i−1
>
when
m1 = 1
, and we have
ai,i−1
≥
j=1
ai,j when
m1
> 1
,
Now, we prove that the common zeros of the system
F1 =
· · · =
Fn−1 = 0
is nothing but
Cm1
,m2
,.,mn. If
x0 = 0
, F1 = 0 yields
x1 = 0
, and thus we have
x2 =
· · · =
xn−1 = 0 by
F2 =
· · · =
Fn−1 = 0
. Thus, the common solution isjust the point (0 :
. . . : 0 : 1) which is on the curve
Cm1
,m2
,.,mn. On the otherhand, we can set
x0 = 1 when
x0 = 0
. Therefore, it is sufficient to show that
the only common solution of these equations is
xi =
tmi, for some
t ∈ K andfor all 1
≤ i ≤ n, which we prove by induction on
i. More precisely, we shownext that if
Fi−1(
x0
, . . . , xn) = 0 and
x0 = 1
, x1 =
tm1
, . . . ,
xi−1 =
tmi−1
,then
xi =
tmi, for all 2
≤ i ≤ n.
By
mi =
ai,i−1
mi−1
−
j=1
ai,j mj, we get gcd(
m1
, . . . , mi−1) = 1 for all
3
≤ i ≤ n. In particular, gcd(
m1
, m2) = 1
, which means that there are integers
1
, 2 such that 1 is positive and 1
m2 + 2
m1 = 1
. From the first equation
2
. Letting
x1 =
T m1
, we get
x2 =
εT m2
, where
ε is an
m1-th root of unity. Setting
t =
ε 1
T, we obtain
x1 =
tm1 and
x2 =
tm2
, whichcompletes the base statement for the induction.
Now, we assume that
x0 = 1
, x1 =
tm1
, . . . , xi−1 =
tmi−1 for some 3
≤ i ≤ n.
Substituting these to the equation
Fi−1 = 0
, we get
i (
tmi−1 )
mi−kai,i−1
Since
mi =
ai,i−1
mi−1
−
i (
tmi )
mi−1
−k = (
xi − tmi )
mi−1 = 0
.
The first direct consequence of our main result is stated below which recov-
ers the main result in when
p = 1
.
Corollary 2.2. If r =
aq − bp for some nonnegative integers a, b such that
a > b when p = 1
and a ≥ bp when p > 1
, then the monomial curve Cp,q,r is
a set-theoretic complete intersection of two surfaces with equations
The following example illustrates the strength of this corollary.
Example 2.3. We consider the monomial curve
C2
,3
,r, r ≥ 4
. Using the results
in ] one may only prove that
C2
,3
,4 is an s.t.c.i. We will prove that
C2
,3
,r are
all s.t.c.i, for
r ≥ 4 except
r = 5 which is addressed in
Clearly,
r = 4
c, r = 4
c + 1
, r = 4
c + 2 or
r = 4
c + 3 for some
c ≥ 1
. For
r = 4
c:
a = 2
c and
b =
c; for
r = 4
c + 2:
a = 2
c and
b =
c − 1 ; for
r = 4
c + 3:
a = 2
c + 1 and
b =
c satisfy the conditions of Corollary and thus the claimfollows. For the case of
r = 4
c + 1
, c ≥ 2
, we have three different situations.
When
c = 3
d + 2
, d ≥ 0:
a = 4
d + 3 and
b = 0; when
c = 3
d, d ≥ 1:
a = 4
d + 1and
b = 1; and when
c = 3
d + 1
, d ≥ 1:
a = 4
d + 3 and
b = 2 satisfy theconditions of Corollary and hence the claim follows.
Under some mild conditions on the greatest common divisor, our main
result can be made more effective to construct infinitely many s.t.c.i. mono-mial curves in arbitrary dimension.
Proposition 2.4. Assume for each integer 3
≤ i ≤ n that there exist an integer
1
≤ ki ≤ i − 2
with gcd(
mki, mi−1) = 1
and that mi ≥ mkimi−1(
mi−1
− 1)
,
then the monomial curve Cm
is a set-theoretic complete intersection.
Proof. For each 3
≤ i ≤ n, from the condition gcd(
mki, mi−1) = 1
, there existpositive integers
Ai and
Bi, such that
mi =
Aimk −
i mi−1(
mi−1
− 1)
≤ mi, mki mi−1(
mi−1
− 1)
≤ Aimki
mkimi−1(
mi−1
− 1) +
Bimi−1
≤ Aimki. Subtracting
Aimkimi−1 from bothhand sides and rearranging, we obtain
i mi−1 +
mki mi−1(
mi−1
− 1)
≤ −Aimki mi−1
− 1)
.
Bi − Aimki + 1
≤ − Ai .
Therefore, we can chose integers
θi such that
Let us now set
ai,i−1 =
−Bi − mk
Ai − mi−1
θi. It follows
then that
mi =
ai,i−1
mi−1
− ai,k
ai,i−1
> 0
, ai,ki
i mki . From Theorem the monomial curve
Cm1
,m2
,.,mn
Using Proposition one can produce infinitely many s.t.c.i. monomial
curves in projective 4
−space as the following example illustrates.
Example 2.5. We consider the monomial curve
C2
,3
,13
, . Since gcd(2
, 3) = 1
,
gcd(2
, 13) = 1
, and 13
≥ 2
.3
.2
, by Proposition the monomial curve
C2
,3
,13
,
is an s.t.c.i. for each
≥ 2
.13
.12 = 312
.
Specializing to monomial space curves, we directly get the following:
Corollary 2.6. If gcd(
p, q) = 1
and r ≥ pq(
q − 1)
, then the monomial curve
Cp,q,r is a set-theoretic complete intersection.
Remark 2.7. From Corollary the monomial curve
Cp,q,(
p+
q)
s is a set-the-
oretic complete intersection for all
s ≥ 3
. Our results do not apply when
s = 1
or
s = 2
. When
Cp,q,p+
q is arithmetically Cohen–Macaulay it is shown to be
s.t.c.i. in ]. Thus the question of whether
Cp,q,(
p+
q)
s is an s.t.c.i. is still
open for
s = 1 and
s = 2
.
Acknowledgements. The author would like to thank Mesut S
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Department of Mathematics,Dong Thap University,Dong Thap,Vietname-mail:
[email protected]
Source: ftp://file.viasm.org/Hotro/ChuongtrinhToan/Thuongcongtrinh/ThuongCongTrinh-2013/2013_Bai_089.pdf
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