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**Archiv der Mathematik**
**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

*m*1

*< · · · < mn *be some positive integers with

gcd(

*m*1

*, . . . , mn*) = 1

*. *A monomial curve

*Cm*
P

*n *over an algebraically closed ﬁeld

*K *is a curve with generic zero
(

*umn, umn−m*1

*vm*1

*, . . . , 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

*f*1

*, . . . , fn−*1 if it is the inter-section of (

*n − *1) hypersurfaces deﬁned by nonzero homogeneous polynomials

*f*1

*, . . . , fn−*1 in a polynomial ring over

*K*; that is, if we can write

*Cm*
*Z*(

*f*1

*, . . . , 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

*m*1

*, . . . , 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 m*1

*< · · · < mn be some positive integers with the prop-*

erty gcd(

*m*1

*, . . . , mn*) = 1

*and satisfying the following two conditions for some*

nonnegative integers ai,j:
(I)

*mi *=

*ai,i−*1

*mi−*1

*−*
*m*1 = 1 and

*ai,i−*1

*>*
*m*1

*> *1 and

*ai,i−*1

*≥*
*j*=1

*ai,j mj, *for all 3

*≤ i ≤ n. *Then the monomial curve

*Cm*1

*,.,mn*
is a set-theoretic complete intersection on

*F*1

*, . . . , Fn−*1

*, *where

*F*1 =

*xm*2

*, *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

*x*0

*, *we have

*ai,i−*1

*>*
when

*m*1 = 1

*, *and we have

*ai,i−*1

*≥*
*j*=1

*ai,j *when

*m*1

*> *1

*,*
Now, we prove that the common zeros of the system

*F*1 =

*· · · *=

*Fn−*1 = 0
is nothing but

*Cm*1

*,m*2

*,.,mn. *If

*x*0 = 0

*, F*1 = 0 yields

*x*1 = 0

*, *and thus we have

*x*2 =

*· · · *=

*xn−*1 = 0 by

*F*2 =

*· · · *=

*Fn−*1 = 0

*. *Thus, the common solution isjust the point (0 :

*. . . *: 0 : 1) which is on the curve

*Cm*1

*,m*2

*,.,mn. *On the otherhand, we can set

*x*0 = 1 when

*x*0 = 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(

*x*0

*, . . . , xn*) = 0 and

*x*0 = 1

*, x*1 =

*tm*1

*, *. . . ,

*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(

*m*1

*, . . . , mi−*1) = 1 for all
3

*≤ i ≤ n. *In particular, gcd(

*m*1

*, m*2) = 1

*, *which means that there are integers
1

*, *2 such that 1 is positive and 1

*m*2 + 2

*m*1 = 1

*. *From the ﬁrst equation
2

*. *Letting

*x*1 =

*T m*1

*, *we get

*x*2 =

*εT m*2

*, *where

*ε *is an

*m*1-th root of unity. Setting

*t *=

*ε *1

*T, *we obtain

*x*1 =

*tm*1 and

*x*2 =

*tm*2

*, *whichcompletes the base statement for the induction.

Now, we assume that

*x*0 = 1

*, x*1 =

*tm*1

*, . . . , 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

*C*2

*,*3

*,r, r ≥ *4

*. *Using the results

in ] one may only prove that

*C*2

*,*3

*,*4 is an s.t.c.i. We will prove that

*C*2

*,*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

*Cm*1

*,m*2

*,.,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

*C*2

*,*3

*,*13

*, . *Since gcd(2

*, *3) = 1

*,*

gcd(2

*, *13) = 1

*, *and 13

*≥ *2

*.*3

*.*2

*, *by Proposition the monomial curve

*C*2

*,*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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