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Archiv der Mathematik
Set-theoretic complete intersection monomial curves in Pn
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
Pn over an algebraically closed field K is a curve with generic zero (umn, umn−m1vm1, . . . , umn−mn−1vmn−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 polynomialsf1, . . . , 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 Pn 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 Pn 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 Pn starting with an s.t.c.i. monomial curve in Pn−1,see e.g. ], it is rather difficult to prove that certain families of monomialcurves are s.t.c.i. in Pn 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 Pn 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−1mi−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 havex2 = · · · = 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−1mi−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 1m2 + 2m1 = 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 1T, 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−1mi−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 = 4c, r = 4c + 1, r = 4c + 2 or r = 4c + 3 for some c ≥ 1. For r = 4c: a = 2c and b = c; for r = 4c + 2: a = 2c and b = c − 1 ; for r = 4c + 3:a = 2c + 1 and b = c satisfy the conditions of Corollary and thus the claimfollows. For the case of r = 4c + 1, c ≥ 2, we have three different situations.
When c = 3d + 2, d ≥ 0: a = 4d + 3 and b = 0; when c = 3d, d ≥ 1: a = 4d + 1and b = 1; and when c = 3d + 1, d ≥ 1: a = 4d + 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−1θi. It follows then that mi = ai,i−1mi−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]

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