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 CmZ(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 xProof. 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) ≤ Aimkimkimi−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 − mkAi − mi−1θi. It follows
then that mi = ai,i−1mi−1 − ai,kai,i−1 > 0, ai,kii 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 References
[1] M. Barile, M. Morales, and A. Thoma, Set-theoretic complete intersections
on binomials, Proc. Amer. Math. Soc. 130 (2001), 1893–1903.
[2] H. Bresinsky, Monomial Gorenstein Curves in A4 as Set-Theoretic Complete
Intersections, Manuscripta math. 27 (1979), 353–358.
[3] H. Bresinsky, Monomial space curves in A3 as set theoretic complete intersec-
tions, Proc. Amer. Math. Soc. 75 (1979), 23–24.
[4] R. C. Cowsik and M. V. Nori, Aﬃne Curves in Characteristic p are Set
Theoretic Complete Intersections, Inventiones Math. 45 (1978), 111–114.
[5] H. Bresinsky, P. Schenzel, and W. Vogel, On liaison, arithmetical Buchs-
baum curves and monomial curves in P3, Journal of Algebra 86 (1984), 283–301.
[6] R. Hartshorne, Complete intersections in characteristic p > 0, Amer. J. Math. 101 (1979), 380–383.
[7] K. Eto, Set-theoretic complete intersection lattice ideals in monoid rings, Jour-
nal of Algebra 299 (2006), 689–706.
[8] J. H. Keum, Monomial curves which are set-theoretic complete intersections,
Comm. Korean Math. Soc. 11 (1996), 627–631.
[9] T. T. Moh, Set-theoretic complete intersections, Proc. Amer. Math. Soc. 94
[10] M. Morales, Noetherian Symbolic Blow-Ups, Journal of Algebra 140 (1991),
[11] L. Robbiano and G. Valla, On set-theoretic complete intersections in the
projective space, Rend. Sem. Mat. Fis. Milano LIII (1983), 333–346.
[12] L. Robbiano and G. Valla, Some curves in P3 are set-theoretic complete
intersections, in: Algebraic Geometry-Open problems, Proceedings Ravello 1982,Lecture Notes in Mathematics, Vol 997 (Springer, New York, 1983), 391–346.
¸ ahin, On Symmetric Monomial curves in P3, Turkish J. Math. 32 (2008),
¸ ahin, Producing set-theoretic complete intersection monomial curves in Pn,
Proc. Amer. Math. Soc. 137 (2009), 1223–1233.
[15] A. Thoma, Monomial space curves in P3k as binomial set-theoretic complete
intersection, Proc. Amer. Math. Soc. 107 (1989) 55–61.
[16] A. Thoma, On the set-theoretic complete intersection problem for monomial
curves in An and Pn, Journal of Pure and Applied Algebra 104 (1995), 333–344.
Department of Mathematics,Dong Thap University,Dong Thap,Vietname-mail: tranhoaingocnhan@gmail.com

Neuigkeiten von Arzneimittel in der Palliativmedizin APM Arzneimittel in der Pal iativmedizin . 5 Wil kommen zur März/April 2012 Ausgabe des APM Newsletter. Sie finden hier Informationen zu aktuel en Themen im Bereich der pal iativmedizinschen Arzneimitteltherapie. Der Newsletter ist Teil des Angebotes, das Ihnen kostenfrei unter www.arzneimittel-pal iativ.de zur Verfügung steht. Für das

Inappropriate Medication Prescribing in Residential Care/Assisted Living Facilities Philip D. Sloane, MD, MPH,* Sheryl Zimmerman, PhD,* Lori C. Brown, PharmD,‡Timothy J. Ives, PharmD, MPH,† and Joan F. Walsh, PhD* OBJECTIVES: To identify the extent to which inappro- estimating equations, IPM use was associated with thepriately prescribed medications (IPMs) are administered tonumber o