On Simplicial Toric Varieties Which Are Set-Theoretic Complete Intersections

doi:10.1006rjabr.1999.8195, available online at http:rrwww.idealibrary.com on

On Simplicial Toric Varieties Which Are Set-Theoretic

Complete Intersections

Margherita Barile

Dipartimento di Matematica, Uni¨ersita degli Studi di Bari, Via Orabona 4,`

70125 Bari, Italy

Marcel Morales

Uni¨ersite de Grenoble I, Institut Fourier, URA 188, B.P.74, 38402 Saint-Martin´

D’Heres Cedex, France; and IUFM de Lyon, 5 rue Anselme, 69317 Lyon Cedex, France`

and Apostolos Thoma

Department of Mathematics, Uni¨ersity of Ioannina, Ioannina 45110, Greece Communicated by Craig Huneke

Received July 22, 1999

In this paper we prove:

1. In characteristic p)0 every simplicial toric affine or projective variety with full parametrization is a set-theoretic complete intersection. This extends

Ž .

previous results by R. Hartshorne 1979, Amer. J. Math.101, 380᎐383 and T. T.

Ž .

Moh 1985,Proc. Amer.Math.Soc.94, 217᎐220 .

2. In any characteristic, every simplicial toric affine or projective variety with full parametrization is an almost set-theoretic complete intersection. This extends

Ž

previous known results by M. Barile and M. Morales 1998,Comm. Algebra26,

. Ž .

1907᎐1912 and A. Thoma Arch. Math., to appear .

3. In any characteristic, every simplicial toric affine or projective variety of codimension two is an almost set-theoretic complete intersection.

Moreover the proofs are constructive and the equations we find are binomial ones.

䊚2000 Academic Press

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INTRODUCTION

An important problem in algebraic geometry is to determine the mini-mum number of equations needed to define an algebraic variety V

Ž .

set-theoretically: if IsI V is the defining ideal of V, this number is Ž .

called the arithmetical rank of I and is denoted ara I . In this paper we only consider ideals generated by binomials. It is natural to define the

Ž Ž ..

binomial arithmetical rank of a binomial ideal I written bar I as the smallest integer sfor which there exist binomials f1, . . . ,fs in I such that

Ž . Ž .

rad I srad f1, . . . ,fs. Hence the binomial arithmetical rank is an upper bound for the arithmetical rank of a binomial ideal. From the definitions we deduce the following inequality for a binomial ideal I:

h I

Ž .

Fara

Ž .

I Fbar

Ž .

I F␮

Ž .

I .

Ž . Ž .

Here h I denotes the height and ␮ I denotes the minimal number of

Ž . Ž . Ž

generators of I. When h I sara I the ideal I and the variety V as

. Ž . Ž . Ž .

well is called a set-theoretic complete intersection s.t.c.i. ; when h I s␮ I

it is called a complete intersection. The ideal I is called an almost set-theo -Ž . Ž .

retic complete intersection if ara I Fh I q1. The binomial arithmetical rank was computed for the defining ideals of monomial curves inPn _{in a}

K Ž w x.

series of articles see 1, 4, 8 . Here is a summary of the results. Let C be a monomial curve in Pn_.

K

Ž .i If the characteristic of K is positive, then barŽ ŽI C..sny1. Ž .ii If the characteristic of K is zero, then barŽ ŽI C..sny1 if C is

Ž Ž ..

a complete intersection and bar I C sn, otherwise. In this article we extend these results and we prove that:

1. In characteristic p)0 any simplicial toric affine or projective variety with full parametrization is a set-theoretic complete intersection.

2. In any characteristic, any simplicial toric affine or projective variety with full parametrization is an almost set-theoretic complete inter-section.

3. In any characteristic, every simplicial toric affine or projective variety of codimension two is an almost set-theoretic complete intersec-tion.

In the sequel we shall use the following notation: Let K be a field. A

parametrization x sud1 1 1 . . . x sudn n n y sua1 , 1 ⭈⭈⭈ _ua1 ,n 1 1 n . . . y suar, 1 ⭈⭈⭈ _uar,n r 1 n

for some positive integers d₁, . . . ,d_n and some integers a_i,j, where, for all

is1, . . . ,r, at least one of a_{i, 1}, . . . ,a_i,n is non-zero. Here we refer to the

w x

definition of toric variety given in 7 , which also includes non-normal varieties. The toric variety V is called simplicial if all the exponents are

Ž .

nonnegative. Let IsI V be the ideal formed by the polynomials of

w x

K x1, . . . ,xn,y1, . . . ,yr vanishing on V. We shall refer to I as to the defining ideal of V. The ideal I has a system of generators formed by binomials which are differences of two monomials with coefficient 1.

w x

A proof is given in 7 .

1. SIMPLICIAL TORIC VARIETIES WITH FULL SUPPORT ARE S.T.C.I. IN CHARACTERISTIC p)0

1.1. General Results

We refer to the variety V and its parametrization introduced above. Let

⌽:⺪rª⺪rd₁⺪=⭈⭈⭈=⺪rd_n⺪

be the homomorphism of groups defined by

w

x

s , . . . ,s ¬ s a qs a q⭈⭈⭈ys a ,

Ž

1 r

.

Ž

1 1 , 1 2 2 , 1 r r, 1

w

x

. . . , s a1 1 ,nqs a2 2 ,nq⭈⭈⭈ys ar r,n

.

. The elements of the lattice

s , . . . ,s gKer⌽

are in a one-to-one correspondence with the binomials of I. Moreover Ker⌽admits a basis of the form

s , 0, . . . , 0 , s ,s , 0, . . . , 0 ,

Ž

Žy1 , 1.

. Ž

Ž0 , 1. Ž0 , 2.

.

. . . ,

Ž

sŽry2 , 1.,sŽry2 , 2., . . . ,sŽry2 ,r.

.

4

.

w x

These are simple generalizations of 5, Remark 2.1.2 . For the sake of

ª _{Ž . Ž .}

simplicity we shall put t0ssŽry2,r., ss s1, . . . ,sry1 , ys y1, . . . ,yry1 .

ª

Ž .

In particular, if s,t gKer⌽, then tgt0⺪and, conversely, for all

multi-ª ry1 ª

Ž .

ples t of t0 there is sg⺪ such that s,t gKer⌽.

ª ry1 ª ª

Remark1. For all sg⺪ let s_qdenote the positive part and s_y the

ª ª

Ž .

negative part of s. Fix an element s,sr gKer⌽, and let

s a1 1 ,iqs a2 2 ,iq⭈⭈⭈ys ar r,is¨idi,

ª ª

for all is1, . . . ,n. Let _¨q denote the positive part and _¨y the negative

ª

Ž .

part of ¨₁, . . . ,¨_n . The binomial corresponding to s is then

ª_s ª_{¨ s} ª_s ª_¨ q y r y q y x yy y xr , if s_rG0; otherwise it is ª_{s ys} ª_¨ ª_s ª_¨ q r y y q y yr x yy x . Let

w

x

JsIlK x1, . . . ,xn,y1, . . . ,yry1 .

Then J is the defining ideal of the simplicial toric variety of codimension

ry1 having the following parametrization:

x sud1 1 1 . . . x sudn n n y sua1 , 1 ⭈⭈⭈ _ua1 ,n 1 1 n . . . y suary1 , 1 ⭈⭈⭈ _uary1 ,n_. ry1 1 n

We introduce one more piece of notation. Let M₁, M₂ be monomials, and let hsM₁yM₂. For all positive integers q we set

hŽq.s_Mq_yMq_.

1 2

LEMMA1. Let␦)0 be an integer for which there is a binomial

ª t0␦ s␦ l1 ln f_rsy_r yy x₁ ⭈⭈⭈ x_n gI. Then hŽ␦.g _J,f

Ž

r

.

for all binomials h in I.

Proof. Let hgI be a binomial. Since I is a prime ideal, we may assume that

hsyt₀␳_{g yg}

r 1 2

w x

for some monomials g1,g2gK x1, . . . ,xn,y1, . . . ,yry1. Then

hŽ␦.s_yt0␳␦_g␦y_g␦ r 1 2 ␳ ª Ž␳. s␦ l1 ln ␦ ␦ s

ž

fr q

_ž

y x1 ⭈⭈⭈ xn

_/

/

g1yg2 g

Ž

J,fr

.

. 1.2. Full Parametrization

We say that the above parametrization of V is full if ai,j/0 for all Ži,j.. In this case the parametrization of the variety defined by J is full, too.

LEMMA2. For all sufficiently large integers␦)0 there is a binomial

ª

t₀␦ s l₁ l_n frsyr yy x1 ⭈⭈⭈ xn gI.

6 6

Ž .

Proof. Let ␦)0. There is s⬘ such that s⬘,t₀ gKer⌽. There are also some integers r₁X, . . . ,rX_n for which

ry1

X X

s a yt a sr d

Ý

j j,i 0 r,i i i

for all i. Multiplying this relation by ␦)0 we obtain ry1 X X ␦s a yt ␦a s␦r d

Ý

j j,i 0 r,i i i js1 4 X

for all i. Let dslcm d1, . . . ,dn ; then up to replacing ␦sj with its residue modulo d, for all i we get a relation

ry1

s a yt ␦a sr d,

Ý

j j,i 0 r,i i i j

where 0Fsj-d for all j. Thus, if ␦ is sufficiently large, we will have

ri-0 for all i. But then

ª

t₀␦ s yr₁ yr_n f_rsy_r yy x₁ ⭈⭈⭈ x_n gI

is the binomial required.

As an immediate consequence we have:

COROLLARY1. Let p be a prime number. For all sufficiently large integers m there is a binomial

ª m

t p0 sm l1 ln frsyr yy x1 ⭈⭈⭈ xn gI.

THEOREM 1. Suppose that charKsp)0. Then e_¨ery simplicial toric

¨ariety ha_¨ing a full parametrization is a set-theoretic complete intersection.

Proof. We proceed by induction on rG1. Since the polynomial ring

w x

K x₁, . . . ,x_n,y₁ is an UFD the claim is true for rs1.

Suppose that rG2 and the claim is true in codimension ry1. Let

hgI be a binomial; then by Corollary 1 and Lemma 1 we get

hpmshŽpm.g _{f ,J}

Ž

r

.

formsufficiently large. By the inductive hypothesis the ideal J is set-theo-retically generated by ry1 binomials f₁, . . . ,f_ry1. Hence some power of

Ž .

h lies in f₁, . . . ,fr .

Remark 2. Note that the proof of the preceding result yields a recur-sive construction of the defining equations of the simplicial toric variety for any field K of characteristic p)0.

2. ALMOST SET-THEORETIC COMPLETE INTERSECTIONS

In this section we show that simplicial toric varieties having a full parametrization are almost set-theoretic complete intersections.

With respect to the notation introduced above, for all is1, . . . ,r let

d₁ 0 ⭈⭈⭈ 0 a_{1 , 1} ⭈⭈⭈ a_{i, 1} 0 d₂ ⭈⭈⭈ 0 a_{1 , 2} ⭈⭈⭈ a_{i, 2} Ais . ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈ ⭈⭈⭈

0 0 ⭈⭈⭈ d_n a_{1 ,n} ⭈⭈⭈ a_i,n

0

w x

Moreover, let D j1, . . . ,jn be the determinant of the n=n submatrix

4

consisting of the columns of A_rwith the indices j₁, . . . ,j_n, where j₁, . . . ,j_n

4 < <

is an n-subset of 1, 2, . . . ,nqr . For all is1, . . . ,r let A_i [

w x 4

gcd D j₁, . . . ,j_n : 1Fj₁-j₂- ⭈⭈⭈ -j_nFnqi; for the sake of

simplic-< <

ity we set g_is A_i. Moreover, let e_isg_iy1rg_i, for all is2, . . . ,r.

4

LEMMA3. Let V be a simplicial toric¨ariety.Then for e¨ery ig 1, . . . ,r there exist binomials

M yN yeig_{I V}

_Ž

_.

_,

i i i

w x

where M_i, N_i are monomials in K x₁, . . . , x_n, y₁, . . . , y_iy1 . If the parametrization of V is full, then for e¨ery is2, . . . ,r there exists a binomial

Fsy␮i y_x␯i, 1 ⭈⭈⭈ _x␯i,n_y␮i, 1 ⭈⭈⭈ _y␮i,iy2_yeig_{I V}

_Ž

_.

_,

i iy1 1 n 1 iy2 i

and there also exists a binomial

F sye1y_x␯1 , 1 ⭈⭈⭈ _x␯1 ,ng_{I V}

_Ž

_.

_,

1 1 1 n

for some positi_¨e integers␮_i,j and␯_i,j.

Proof. In this proofd_i will denote the ith column vector of A_r for all

Ž .

is1, . . . ,n, and a_i will denote the nqi th column vector of A_r for

Ž . Ž .

all is1, . . . ,r. Set ␮slcm d₁, . . . ,dn and qisgcd ␮,ai, 1, . . . ,ai,n for all is1, . . . ,r. For all is1, . . . ,r and all js1, . . . ,n let ␳i,js ai,j␮rd qj j. Then, for all is1, . . . ,r, one has that

G sy␮rq_iyx␳i, 1 ⭈⭈⭈ x␳i,ngI V

_Ž

_.

_.

i i 1 n

It is easy to see that e1s␮rq1; then for is1 the preceding formula yields the required binomial F1.

Ž w x.

By a basic lemma in number theory see 3 the diophantine system

< < < < < <

Axsb has a solution iff A /0 and A s Ab, where Ab is the augmented matrix.

w x

Let 2FiFr. The integer giy1 is a divisor of D1, . . . ,n sd d0 1 ⭈⭈⭈ dn

/0; hence giy1/0. On the other hand the following hold:

w

x

gisgcd

giy1,D j1, . . . ,jny1,nqi : 1Fj1-j2- ⭈⭈⭈ -jnFnqiy1 ,

4

Ž .

1 <Aiy1,eiai<sgcd

giy1,

Ž

giy1rg D ji

.

w

1, . . . ,jny1,nqi

x

: 1FjkFnqiy1

4

w

x

s

Ž

giy1rgi

.

gcd

gi,D j1, . . . ,jny1,nqi : 1FjkFnqiy1

4

sgiy1. Hence the diophantine system Aiy1xseiai always has a solution. This means that the vectoreiaican be expressed as a linear combination of the vectorsd1, . . . ,dn,a1, . . . ,aiy1 with integer coefficients; i.e., one has

eiaist1d1q⭈⭈⭈qtndnqtnq1a1q⭈⭈⭈qtnqiy1aiy1,

Ž .

2 for some integers t1, . . . ,tnqiy1. This expression gives us monomials Mi,Ni

w x e_i

Ž . in K x1, . . . ,xn,y1, . . . ,yiy1 such that MiyN yi i gI V .

Now suppose that the parametrization of V is full. From the binomial

Gj we see that for each aj there exist positive integers ␳js␮rqi,

␳j, 1, . . . ,␳j,n such that ␳jajs␳j, 1d1q⭈⭈⭈q␳j,ndn. Furthermore, for all 1FjFiy2, there exists a positive integer ␯j such that, after adding all

Ž .

the zero vectors ␯ ␳j j, 1d1q⭈⭈⭈q␳j,ndny␳jaj to the right-hand side of Ž .2 , the new coefficient ␮_i,k of a_k is negative for all ks1, . . . ,iy2. There also exists a large positive integer ␯_iy1 such that after adding the

Ž Ž ..

zero vector␯_iy1 ␳_iy1a_iy1y ␳_{iy1, 1}d₁y⭈⭈⭈y␳_iy1,nd_n on the right-hand side of the new equation, for all js1, . . . ,nthe new coefficient ␯_i,j ofd_j

is negative and the new coefficient ␮_i ofa_i is positive. It follows that for all is2, . . . ,r

Fsy␮i y_x␯i, 1 ⭈⭈⭈ _x␯i,n_y␮i, 1 ⭈⭈⭈ _y␮i,iy2_yeig_{I V}

_Ž

_.

_.

i iy1 1 n 1 iy2 i

THEOREM2. Let V be a simplicial toric¨ariety ha¨ing a full parametriza -Ž -Ž ..

tion.Then rFbar I V Frq1.

Proof. Consider the r binomials F₁,F₂, . . . ,Fr which were defined in Lemma 3 and let Frq1 be any binomial monic in yr, for example, Gr. We

Ž . Ž .

claim that I V srad F1, . . . ,Frq1 . By virtue of Hilbert Nullstellensatz the claim is proved once it has been shown that every point xs

nqr Žx₁, . . . ,x_n,y₁, . . . ,y_r. which is a common zero of F₁, . . . ,F_rq1 in K , where K denotes the algebraic closure of K, is also a point of V. First of all note that if xks0 for some index k, then yjs0 for all indices j. It is then easy to find u₁, . . . ,ungK which allows us to writexas a point of V. Now suppose that xk/0 for all indices k. By induction on i, 2FiFrq1, we show: ifx is a zero of F1, . . . ,Fiy1, then the coordinates ofxfulfill the parametrization of V. The claim is easy for is2. Now fix an index i, 2FiFrq1. By the induction hypothesis there are nonzero u1, . . . ,ung K such that

x sud1_{, . . . ,x} s_udn_{, y} s_ua1 , 1 ⭈⭈⭈ _ua1 ,n_{, . . . ,y} s_uaiy1 , 1 ⭈⭈⭈ _uaiy1 ,n_.

1 1 n n 1 1 n iy1 1 n

Since the pointx is also a zero of F, we deduce that y s␻uai, 1 ⭈⭈⭈ _uai,n_,

i i 1 n

gi

where ␻ is a suitable ei-root of unity. Let ␨gK be such that ␨ s␻, so

giy1 _{Ž .}

that ␨ s1. By 1 and Bezout’s identity there exist integers

´

k0 and

kj₁,⭈⭈⭈,j_ny1,nqi such that

w

x

g_isk g_{0 iy1}q

_Ý

k_{j,⭈⭈⭈,j ,nqi}D j₁, . . . ,j_ny1,nqi . 1 ny1

w x

All the D j1, . . . ,jny1,nqi are linear combinations of ai, 1, . . . ,ai,n. Therefore there exist l1, . . . ,ln such that gisk g0 iy1ql a1 i, 1q⭈⭈⭈ql an i,n. Setting ¨ s␨lju_{, we have that}

j j x s¨d1_{, . . . ,x} s¨dn_{, y} s¨a1 , 1 ⭈⭈⭈ ¨a1 ,n_{, . . . ,y} s¨aiy1 , 1 ⭈⭈⭈¨aiy1 ,n_, 1 1 n n 1 1 n iy1 1 n y s¨ai, 1 ⭈⭈⭈¨ai,n_, i 1 n since ␻s␨g_is␨k₀g_iy1ql₁a_{i, 1}q⭈⭈⭈ql_na_i,ns␨l₁a_{i, 1}q⭈⭈⭈ql_na_i,n and also 1s␨ljdj_{, since}

w

x

l dj jsl1=0q⭈⭈⭈ql dj jq⭈⭈⭈qln=0s

_Ý

kj1,⭈⭈⭈,jny1,nqiD j1, . . . ,jny1,j . Moreover, 1s␨l1af, 1q⭈⭈⭈qlnaf,n _{for f}-_{i. In fact one has that}

w

x

l a_{1 f, 1}q⭈⭈⭈ql a_{n f,n}s

_Ý

k_{j⭈⭈⭈,,j ,nqi}D j₁, . . . ,j_ny1,fqn 1 ny1

and one of the following two cases occurs: either fqn is one of the

w x

j1, . . . ,jny1, and D j1, . . . ,jny1,fqn s0, or fqn is different from

w x

j1, . . . ,jny1, and D j1, . . . ,jny1,fqn is a multiple of giy1, since then

w x

D j1, . . . ,jny1,fqn is a subdeterminant of Aiy1. We have shown that the coordinates of the pointxfulfill the parametrization of V.

3. RESULTS IN ARBITRARY CHARACTERISTIC AND CODIMENSION 2

We show that Theorem 2 can be generalized: it can be extended to the varieties which do not have a full parametrization, at least in codimen-sion 2.

In this section we suppose that rs2; i.e.,V is a simplicial toric variety of codimension 2 in Knq2_{. The parametrization of V now is}

x sud1 1 1 ._. . x sud_n n n y sua_{1 , 1 ⭈⭈⭈ u}a_{1 ,}n 1 1 n y sua2 , 1 ⭈⭈⭈ _ua2 ,n_, 2 1 n

where the vectorsa1,a2 may have zero components.

THEOREM 3. Let V be a simplicial toric_¨ariety of codimension 2. Then

Ž Ž .. 2Fbar I V F3.

Proof. Let j₁ be the definining ideal of the simplicial toric variety having the parametrization

x sud1 1 1 . . . x sudn n n y sua_{1 , 1 ⭈⭈⭈ u}a_{1 ,n} , 1 1 n

and let J2 be the defining ideal of the simplicial toric variety having the following parametrization: x sud₁ 1 1 . . . x sudn n n y sua2 , 1 ⭈⭈⭈ _ua2 ,n_. 2 1 n

Both varieties have codimension one; therefore their defining ideals are principal. If we set ␮slcm

Ž

d1, . . . ,dn

.

, q1sgcd

Ž

␮,a1 , 1, . . . ,a1 ,n

.

, e1s␮rq1 and q2sgcd

Ž

␮,a2 , 1, . . . ,a2 ,n

.

, eX1s␮rq2, when J₁ is generated by F sye1y_xa1 , 1␮rd1q1 ⭈⭈⭈ _xa1 ,n␮rdnq1 1 1 1 n and J2 is generated by F syeX1y_xa2 , 1␮rd1q2 ⭈⭈⭈ _xa2 ,n␮rdnq2_. 3 2 1 n

Note that F₁ is the difference of a power of y₁ and a monomial which Ž .

only involves the variables x_k such that kgSuppa₁ ; similarly, F₃ is the difference of a power of y₂ and a monomial which only involves the

Ž . variables xk such that kgSuppa2 .

e2 _{Ž .}

Let F₂sM₁yM y_{2 2} gI V be the binomial given in Lemma 3. We

Ž . Ž .

claim that I V srad F1,F2,F3 .

nq2

Ž .

Let xs x1, . . . ,xn,y1,y2 be a common zero of F1,F2,F3 in K , where K denotes the algebraic closure of K. We show thatxlies onV. If

xk/0 for all indices k, then the claim can be easily proven.

Ž . Now suppose that xks0 for at least one index kgSuppa1 j

Ž .

Suppa2 . One of the following cases occurs.

Ž .i If kgSuppŽa1.lSuppŽa2., then F1Ž .x s0 implies that y1s0, Ž .

and F3x s0 implies that y2s0. For all is1, . . . ,r let uigK be such that udis_{x. These parameters allow us to writex as a point of V.}

i i

Ž .ii If kgSuppŽa1., then, again, F1Ž .x s0 implies that y1s0;

Ž . Ž .

moreover, F3 x s0 implies that x is a point of V J2 , with respect to some u1, . . . ,ungK: the same values of the parameters yield a represen-tation ofxas a point of V.

Žiii. If kgSuppŽa₂., one can proceed as in ii .Ž . Ž .

Remark 3. 1 It is easy to see that if ␣ is an integer for which the equation Aiy1xs␣ai has an integer solution, then ␣ is a multiple of ei. We know from the results in Section 1 that there is a binomial ys0_M y

1 1

t_{0 w x}

y M2 2gI, where M1,M2 are monomials of K x1, . . . ,xn . Hence t0 is a multiple of e2; on the other hand we know that in any binomial of I the exponent of y2 is a multiple of t0. Thus t0se2.

Ž .2 It follows from 5, Theorem 3.5 that the binomialsw x F₁,F₂,F₃

belong to a Grobner basis of

_¨

I.

COROLLARY 2. With respect to the notations introduced abo¨e, one has that

1. If eX spm_{e for some prime p)0, then V is a set-theoretic com}

-1 2

plete intersection in characteristic p.

2. If eX₁se then V is a complete intersection in any characteristic₂ .

Ž . pm Ž .

Proof. 1 It is clear that F₂ g F₁,F₃ . Ž .2 This is immediate from 5, Theorem 3.5 .w x

EXAMPLE. Let V be the simplicial toric variety of codimension 2 parametrized by ass20_{, bst}20_{, csu}20_, ds_¨20 , yst12_u5_¨3 , zss10_t3_¨7 .

Note that the parametrization is not full. The ideal I is minimally gener-ated by

b12_c5_d3_yy20_{, a}8_d5_y4_ycz16_{, a}2_y16_yb9_c4_dz4_,

a2_b3_cd2_yy4_z4_{, a}4_dy12_yb6_c3_z8_,

a6_d3_y8_yb3_c2_z12_{, a}10_b3_d7_yz20_.

With respect to the notations of Theorem 3 one has that

g s204_{, g s20}3_{, g s5=20}2_{, e s20, e s4} 0 1 2 1 2 5=202s

Ž

7=20y10=12y3=5

.

=202 l s y12=202_{, l s y5=20}2_{, l s0, l s20=20}2 1 2 3 4 and Israd b12_c5_d3_yy20_,a2_b3_cd2_yy4_z4_,a10_b3_d7_yz20

Ž

.

in any characteristic different from 5. In characteristic 5 we have that

Israd b12_c5_d3_yy20_,a10_b3_d7_yz20 _.

Ž

.

Remark 4. If V is arithmetically Cohen᎐Macaulay, then according to

w5, Theorem 3.5, 6 , the varietyx V is a s.t.c.i. on a binomial and a polynomial.

REFERENCES

1. M. Barile and M. Morales, On the equations defining projective monomial curves,Comm. Ž .

Algebra26 1998 , 1907᎐1912.

Ž . 2. R. Hartshorne, Complete intersections in characteristic p)0, Amer.J.Math.101 1979 ,

380᎐383.

Ž . 3. J. Heger, Denkschriften, Kais. Akad.Wissensch. Mathem.Naturw.Klasse14 1858 , II.

Ž . 4. T. T. Moh, Set-theoretic complete intersections, Proc. Amer. Math. Soc. 94 1985 ,

217_᎐220.

5. M. Morales, Equations des Varietes Monomiales en codimension deux,´ ´ J. Algebra175

Ž1995 , 1082. ᎐1095.

6. L. Robbiano and G. Valla, Some curves inP3_{are set-theoretic complete intersections,in}

Ž

‘‘Algebraic GeometryᎏOpen Problems, Proc. Ravello 1982’’ Ciliberto, Ghione, Orecchia, .

Eds. , Lecture Notes in Mathematics, Vol. 997, pp. 391᎐399, Springer-Verlag, BerlinrNew York, 1983.

7. B. Sturmfels, ‘‘Grobner Bases and Convex Polytopes,’’ University Lecture Series, No. 8,¨

Am. Math. Soc., Providence, 1995.

Ž . 8. A. Thoma, On the binomial arithmetical rank, Arch.Math.73 1999 , 1᎐4.