aa r X i v : . [ m a t h . A C ] O c t AFFINE MONOMIAL CURVES
INDRANATH SENGUPTAA
BSTRACT . This article is an expository survey on affine monomialcurves, where we discuss some research problems from the perspectiveof computation.
This article is an attempt to introduce and discuss some ongoing researchin the field of affine monomial curves and numerical semigroup rings. Thereis a large body of literature in this area and we would not make any attemptto cite most of them. We would rather cite some, which have a strong list ofreferences, e.g. , [46] and [52]. The topics which we have chosen to discussfor this article are limited and should not be misinterpreted as an exhaus-tive list. We have chosen the topics from the perspective of computation,where computer algebra software like Singular [21] and GAP [18] can beused effectively. This article could be useful for some young researchers,who are interested in the computational aspects of commutative algebra andalgebraic geometry. The texts [34], [2], [44] and [29] provide most of thebackground material in commutative algebra and [22], [16] are strongly rec-ommended for learning the subject using computer algebra as a companion.1. N
UMERICAL S EMIGROUPS AND A FFINE M ONOMIAL C URVES
Let N denote the set of nonnegative integers. A numerical semigroup Γ is a subset of N containing , closed under addition and generates Z as agroup. It follows that (see [46]) the set N \ Γ is finite and that the semigroup Γ has a unique minimal system of generators n < n < · · · < n p , that is, Γ = Γ( n , . . . , n p ) = { p X j =0 z j n j | z j nonnegative integers } . Mathematics Subject Classification.
Primary 00-02; 13P99.
Key words and phrases.
Numerical semigroups, Symmetric numerical semigroups,Ap´ery set, Frobenius number, Minimal presentation, Monomial curves, Gr¨obner basis,Syzygies, Betti numbers, Derivation module, Complete intersection, Francia’s conjecture,Blowup algebras, Set theoretic complete intersection.This author thanks SERB for their support through the MATRICS grantMTR/2018/000420.
The term minimal stands for the following: if n i = P pj =0 z j n j for somenon-negative integers z j , then z j = 0 for all j = i and z i = 1 . The integers n and p + 1 are known as the multiplicity m (Γ) and the embedding dimen-sion e (Γ) . The greatest integer not in Γ is called the Frobenius number of Γ , denoted by F (Γ) . The integer c (Γ) = F (Γ) + 1 is called the condcutor of Γ . The Ap´ery set of Γ with respect to a non-zero a ∈ Γ is defined to bethe set Ap(Γ , a) = { s ∈ Γ | s − a / ∈ Γ } . The numerical semigroup Γ is symmetric if F (Γ) is odd and x ∈ Z \ Γ implies F (Γ) − x ∈ Γ . For ex-ample, if Γ is the numerical semigroup generated minimally by the integers , , , then m (Γ) = 3 , e (Γ) = 3 , F (Γ) = 4 , c (Γ) = 5 . It is easy to verifythat Γ is not a symmetric semigroup. Computing F (Γ) is a hard problemin general, known as the Frobenius Problem (see [1]) or the coin exchangeproblem (see [54]) .Let p ≥ and n , . . . , n p be positive integers with gcd( n , . . . , n p ) = 1 .Let us assume that the numbers n , . . . , n p generate the numerical semi-group Γ( n , . . . , n p ) minimally. Let k denote a field and η : k [ x , . . . , x p ] → k [ t ] be the mapping defined by η ( x i ) = t n i , ≤ i ≤ p . Let p ( n , . . . , n p ) =ker( η ) . The map η is a monomial parametrization and defines a curve C = { ( t n , . . . , t n p ) | t ∈ k } , known as a monomial curve in the affinespace A p +1 . The ideal p ( n , . . . , n p ) is called the defining ideal of thecurve C , which has been called by the name monomial primes by some au-thors; [15], [53]. The affine k -algebra A = k [ x , . . . , x p ] / p ( n , . . . , n p ) = k [ t n , . . . , t n p ] is called the coordinate ring of the affine monomial curve C .Let R denote the polynomial ring k [ x , . . . , x p ] . The ideal p ( n , . . . , n p ) isa prime ideal with dimension one, that is, the Krull dimension of the affine k -algebra A is one, therefore, A is Cohen-Macaulay and p ( n , . . . , n p ) is aperfect ideal.2. D EFINING E QUATIONS , G R ¨ OBNER BASES AND S YZYGIES
The ideal p ( n , . . . , n p ) is a graded ideal with respect to the weighted gra-dation given by wt( x i ) = n i , wt( t ) = 1 on the polynomial rings k [ x , . . . , x p ] and k [ t ] . Therefore, by the graded version of Nakayama’s lemma, the mini-mal number of generators of p ( n , . . . , n p ) is a well-defined notion and thisnumber is denoted by β ( p ( n , . . . , n p )) , known as the first Betti number of p ( n , . . . , n p ) . It is both interesting and hard to compute a minimal gen-erating set and β ( p ( n , . . . , n p )) for the defining ideal p ( n , . . . , n p ) . It isnot difficult to show that the ideal p ( n , . . . , n p ) is generated by binomi-als; see Lemma 4.1 in [55] for a more general statement on toric ideals).Choosing a finite number of them minimally usually requires a good under-standing of the Ap´ery set; see [42], [43], [33], [35]. The computer algebrapackage GAP [18] is quite a useful companion for these calculations. It is FFINE MONOMIAL CURVES 3 usually harder to compute a generating set for p ( n , . . . , n p ) , which formsa Gr¨obner basis. Let us recall the basics of Gr¨obner basis first so that wecan understand its usefulness. We refer to [13] for a good introduction tothe subject.Let k [ x , . . . , x n ] be the polynomial ring over k , with a monomial or-der > . For = f ∈ k [ x , . . . , x n ] , let Lm > ( f ) , Lt > ( f ) , Lc > ( f ) denotethe leading monomial , leading term and leading constant of f respectively.These are often written as Lt( f ) , Lc( f ) , Lm( f ) , omitting “ > ” from theexpression. For = f, g ∈ k [ x , . . . , x n ] , the S - polynomial S ( f, g ) , is thepolynomial S ( f, g ) := lcm (Lm( f ) , Lm( g ))Lt( f ) · f − lcm (Lm( f ) , Lm( g ))Lt( g ) · g . Given G = { g , . . . , g t } ⊆ k [ x , . . . , x n ] and f ∈ k [ x , . . . , x n ] , we saythat f reduces to zero modulo G , denoted by f → G , if f can be writtenas f = P ti =1 a i g i , such that Lm( f ) ≥ Lm( a i g i ) , whenever a i g i = 0 .Buchberger’s Criterion says that, for an ideal I in k [ x , . . . , x n ] generated bythe set G = { g , . . . , g t } , G is a Gr¨obner basis for I iff S ( g i , g j ) → G , forevery i = j . The situation is simpler when gcd( Lm( f ) , Lm( g ) ) = 1 , andone chooses monomial orders in order to maximize the occurrence of suchpairs in the list of polynomials G = { g , . . . , g t } . The following Lemma isoften useful in computing a Gr¨obner basis. Lemma 2.1.
Let G = { g , . . . , g t } ⊆ k [ x , . . . , x n ] and let f, g ∈ G benon-zero with Lc( f ) = Lc( g ) = 1 , and gcd( Lm( f ) , Lm( g ) ) = 1 . Then,(1) S ( f, g ) = Lm( g ) . f − Lm( f ) . g .(2) S ( f, g ) = − ( g − Lm( g )) . f + ( f − Lm( f )) . g −→ G . The computation with the S -polynomials can be used together with theSchreyer’s theorem 2.2 for writing down the first syzygy for an ideal. Werefer to the book by Peeva [44] for syzygies and minimal free resolutions.Let us recall Schreyer’s theorem from Chapter 5, Theorem 3.2 in [14]. Theorem 2.2 (Schreyer) . Let K be a field, k [ x , . . . , x n ] be the polynomialring and I be an ideal in k [ x , . . . , x n ] . Let G := { g , . . . , g t } be an orderedset of generators for I , which is a Gr¨obner basis, with respect to some fixedmonomial order on k [ x , . . . , x n ] . Let Syz( g , . . . , g t ) := { ( a , . . . , a t ) ∈ R t | P ti =1 a i g i = 0 } . Suppose that, for i = j , S ( g i , g j ) = a j g i − a i g j = t X k =1 h k g k −→ G . INDRANATH SENGUPTA
Then, the t -tuples (cid:0) h , · · · , h i − a j , · · · , h j + a i , · · · , h t (cid:1) generate Syz( g , . . . , g t ) . The ideal p ( n , . . . , n p ) is a graded ideal, the R -module A is a gradedmodule with projective dimension p . Therefore, the R -module A has aminimal graded free resolution −→ R β p ϑ p − −→ R β p − −→ · · · −→ R β ϑ −→ R β ϑ −→ R ϑ −→ A −→ . The Betti numbers β i will be denoted by β i ( p ( n , . . . , n p )) , often called the Betti numbers of p ( n , . . . , n p ) . These are important numerical invariantsfor the curve C , particularly, for understanding an embedding of C in A p +1 k .One can see computation of Gr¨obner basis and application of Schreyer’stheorem in the articles [50], [51] and [48]. This is often the only effectiveway to compute a free resolution for the ideal p ( n , . . . , n p ) , with someamount of help from a computer algebra software like Singular [21] inguessing a candidate for the Gr¨obner basis of the ideal. Sometimes, onebecomes lucky when the ideal p ( n , . . . , n p ) has some special structure. Forexample, in the case when n , . . . , n p is a minimal arithmetic sequence, thedefining ideal has a special structure, which allows one to adopt some ho-mological techniques like mapping cone to write a free resolution; see [19],[20]. However, Schreyer’s technique or mapping cone or other techniquesrarely create a free resolution that is minimal. The next difficulty is to spotthe redundancy in the free resolution obtained from one of the techniquesand extract a minimal free resolution which sits as a direct summand ofthe free resolution; see [20], [48]. Some questions emerge from the abovediscussions and let us list them down.Q1. Compute a minimal generating set for the defining ideal p ( n , . . . , n p ) and write a formula for the first Betti number β ( p ( n , . . . , n p )) .Q2. Compute a Gr¨obner basis for the defining ideal p ( n , . . . , n p ) anduse Schreyer’s theorem to compute the first syzygy module. Com-pute a minimal subset of the first syzygy module and write a formulafor the cardinality of that set, known as the second Betti number, de-noted by β ( p ( n , . . . , n p )) .Q3. Write a minimal free resolution for the defining ideal p ( n , . . . , n p ) and compute all the Betti numbers β i ( p ( n , . . . , n p )) . FFINE MONOMIAL CURVES 5
3. B
ETTI NUMBERS AND THEIR UNBOUNDEDNESS
The i -th Betti number β i ( p ( n , . . . , n p )) of the ideal p ( n , . . . , n p ) is therank of the i -th free module in a minimal free resolution of p ( n , . . . , n p ) .For a given p ≥ , let β i ( r ) = sup( β i ( p ( n , . . . , n r ))) , where sup is takenover all the sequences of positive integers n , . . . , n p . Herzog [23] provedthat β (3) = 3 . It is not difficult to show that β (3) is a finite integer as well.Bresinsky in [3], [4], [5], [6], [9], extensively studied relations among thegenerators n , . . . , n p of the numerical semigroup defined by these integers.It was proved in [4] and [5] respectively that, for r = 4 and for certain casesin r = 5 , the symmetry condition on the semigroup generated by n , . . . , n p imposes an upper bound on β ( p ( n , . . . , n p )) . The following question isopen in general:Q4. Given p ≥ , does the symmetry condition on the numerical semi-group minimally generated by n , . . . , n p impose an upper boundon β ( p ( n , . . . , n p )) ?Bresinsky [3] constructed a class of monomial curves in A to prove that β (4) = ∞ . He used this observation to prove that β ( r ) = ∞ , for every r ≥ . Let us recall Bresinsky’s example of monomial curves in A , asdefined in [3]. Let q ≥ be even, q = q + 1 , d = q − . Set n = q q , n = q d , n = q q + d , n = q d . Clearly gcd( n , n , n , n ) =1 . Let us use the shorthand n to denote Bresinsky’s sequence of integers de-fined above. Bresinsky [3] proved that the set A = A ∪ A ∪ { g , g } generates the ideal p ( n , . . . , n ) , where A = { f µ | f µ = x µ − x q − µ − x q − µ x µ +14 , ≤ µ ≤ q } , A = { f | f = x ν x ν − x µ x µ , ν , µ < d } and g = x d − x q , g = x x − x x . We have proved in [35] that forBresinsky’s examples β i (4) = ∞ , for every ≤ i ≤ . We have alsodescribed all the syzygies explicitly. We have in fact proved that the set S = A ∪ A ′ ∪ { g , g } , where A ′ = { h m | x m x ( q − m )4 − x ( q − m )2 x m , ≤ m ≤ q − } is a minimal generating set and also Gr¨obner basis for the ideal p ( n ) with respect to the lexicographic monomial order induced by x >x > x > x on k [ x , . . . , x ] . It turns out that β ( p ( n )) = | S | = 2 q .Then it has been proved using Schreyer’s theorem that β ( p ( n )) = 4( q − and also β ( p ( n )) = 2 q − . A similar study has been carried out by J. Her-zog and D.I. Stamate in [24] and [52]. However, the objective and approachin our study are quite different.The integers n , n , n , n defining Bresinsky’s semigroups have the prop-erty that n + n = n + n , up to a renaming if necessary. We gener-alize this construction in arbitrary embedding dimension by consideringthe family of semigroups Γ( M ) , minimally generated by the set M = INDRANATH SENGUPTA { a, a + d, a + 2 d, . . . , a + ( e − d, b, b + d } , b > a + ( e − d , gcd( a, d ) = 1 and d ∤ ( b − a ) . Note that the sequence of integers is a concatenation of twoarithmetic sequences and the condition d ∤ ( b − a ) ensures that the sequenceis not a part of an arithmetic sequence. See [36] for discussion on this classof semigroups. In [36] the case a = e + 1 has been studies in detail and ithas been proved that µ ( p ) is bounded. In [36] it has been shown that thereare symmetric semigroups of this form, generalizing Roasales’ constructionin [47]. Our observations in [36] give rise to the following question:Q5. Let p ≥ be an integer, b > a + ( p − d , gcd( a, d ) = 1 and d ∤ ( b − a ) . Let M = { a, a + d, a + 2 d, . . . , a + ( p − d, b, b + d } be the set of p + 1 positive integers. Let Γ( M ) = h M i be thenumerical semigroup generated by the set M . We assume that theset M is a minimal system of generators for the semigroup Γ( M ) .The semigroup is known as a semigroup generated by concatanationof two arithmetic sequences. Given p ≥ , what conditions on a, b, d ensure that β i ( p ( M )) are unbounded for every i ?4. T HE D ERIVATION M ODULE
Let k be a field of characteristic zero. We now consider the algebroid caseand denote the coordinate ring of the algebroid monomial curve defined bythe integers n , . . . , n p ; p ≥ , by the same notation A ; A = k [[ x , . . . , x p ]] / p ( n , . . . , n p ) = k [[ t n , . . . , t n p ]] . The module of k -derivations Der k ( A ) , known as the Derivation moduleof A is an extremely important object of study. A good understanding ofthe derivation module is essential for various reasons and surely from theperspective of the Zariski-Lipman conjecture, which says that over a fieldof characteristic , A is a polynomial ring over k if Der k ( A ) is a free A -module; see [25], [39]. An explicit generating set for the derivation modulehas been constructed by Patil and Singh [42] under the assumption that theintegers n , . . . , n p form a minimal almost arithmetic sequence. There arediscussions on other curves in [40] which has interesting results on theirderivation modules. In [49], the minimal number of generators for thederivation modules has been calculated. This also gives a formula for thelast Betti number for the defining ideal p ( n , . . . , n p ) . The following theo-rem was proved by Kraft [31]; page 875, which reduces the computation of µ (Der k ( A )) to a counting problem; see [49]. Theorem 4.1.
The set (cid:8) t α +1 ddt | α ∈ ∆ ′ ∪ { } (cid:9) is a minimal set of gen-erators for the A -module Der k ( A ) , where Γ( n , . . . , n p ) is the numerical FFINE MONOMIAL CURVES 7 semigroup generated by the sequence n , . . . , n p of positive integers, Γ( n , . . . , n p ) + := Γ( n , . . . , n p ) \ { } ;∆ := { α ∈ Z + | α + Γ( n , . . . , n p ) + ⊆ Γ( n , . . . , n p ) } ;∆ ′ := ∆ \ Γ( n , . . . , n p ) . In particular, µ (Der k ( A )) = card (∆ ′ ) + 1 . Q6. Compute µ (Der k ( A )) , where k is a field of characteristic and A = k [[ x , . . . , x p ]] / p ( n , . . . , n p ) .5. S MOOTHNESS OF BLOWUPS AND F RANCIA ’ S CONJECTURE
Let R be a regular local ring. Let p be an ideal in R . The Rees algebra R ( p ) = R [ p T ] is called the Rees Algebra of the ideal p . The BlowupAlgebras, in particular the Rees algebra and the associated graded ring G ( p ) = R ( p ) / p R ( p ) of the ideal p play a crucial role in the birational studyof curves. See [58] for learning the basics and related results on blowup al-gebras. The projective scheme Proj( R ( p )) is the blowup of Spec( R ) along V ( p ) . Francia’s conjecture is the following: Francia’s Conjecture. If p is a prime ideal in a regular local ring R with dim( R/ p ) = 1 and if Proj( R ( p )) is a smooth projective scheme then p is acomplete intersection.The conjecture stated in the context of monomial curves is the following: Francia’s Conjecture for monomial curves. If Proj( R ( p ( n , . . . , n p ))) is a smooth projective scheme, then µ ( p ( n , . . . , n p )) = p , in other words p ( n , . . . , n p ) is a complete intersection.The conjecture is false in general. It was proved in [30], that, if R = Q [ x, y, z ] and if p denotes the ideal generated by the maximal minors of thematrix x − y y − zz − y x − y + zz x − yx y z , then p is a prime ideal of codimension and R [ p t ] is smooth. This producesa counter example for Francia’s conjecture over the base filed k = Q . How-ever, over an algebraically closed field this ideal does not remain a primeideal. Therefore, the conjecture is open in its geometric form, that is, over k = C . The conjecture has been proved for certain cases in [10]. It seemsthat the hardest part in solving this conjecture is computing the equationsdefining the Rees algebra as an affine algebra over R ; see [38]. In [37]we have proved that the conjecture is true if the integers n , . . . , n p form INDRANATH SENGUPTA an arithmetic sequence. We could do this without explicitly computing allthe equations defining the Rees algebra, in order to show that the projectivescheme
Proj( R ( p ( n , . . . , n p ))) is not smooth. This answered the conjec-ture in affirmative because we knew that p ( n , . . . , n p ) could never be acomplete intersection if p ≥ and n , . . . , n p form an arithmetic sequence;see [33]. One case which has not been studied so far is when the integers n , . . . , n p form an almost arithmetic sequence, that is, all but one of theseintegers form an arithmetic sequence.Q 7. Answer Francia’s conjecture when n , . . . , n p form a minimal al-most arithmetic sequence of integers.6. T HE S ET THEORETIC COMPLETE INTERSECTION CONJECTURE
To quote M. Hochster [26], “This seems incredibly difficult”. Let us statethe problem in the context of affine monomial curves.Q6. There exist p polynomials f , . . . , f p in k [ x , . . . , x p ] , such that p ( n , . . . , n p ) = q ( f , . . . , f p ) . The conjecture is true if charactertistic of k is a prime, by the famous resultproved by Cowsik and Nori [12]. However, very little is known in char-acteric zero, except for some special cases. We refer to [32] for a detailedaccount of research done on this conjecture. Patil [41] proved that affinemonomial curves defined by an almost arithmetic sequence is a set theo-retic complete intersection. Bresinsky proved the conjecture in dimension3 in [7] and that in dimension with symmetry condition in [8]. Theseresults were subsequently generalized by Thoma [56]. Eto [17] proved theconjecture when the integers define a balanced semigroup in dimension .The general question is still open.An important technique using Gr¨obner basis was mentioned in [45], whichsays the following: If V is a projective variety of codimension r in P n ,which is the projective closure of an affine variety W in A n , and if I ( W ) = p ( f , . . . , f r ) , such that { f , . . . , f r } is a Gr¨obner basis for the ideal gener-ated by { f , . . . , f r } , then I ( V ) = I ( W ) h = p ( f h , . . . , f hr ) . The authorshave proved that the rational normal curve in P n is a set-theoretic completeintersection by the technique mentioned above. The paper [57] precedes[45] and proves several interesting results on ideals defined by determinan-tal conditions. The set-theoretic complete intersection property of the ratio-nal normal curve forms an important step towards proving the set-theoreticcomplete intersection property of an affine monomial curve defined by analmost arithmetic sequence in [41]. FFINE MONOMIAL CURVES 9
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