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The reduction number and degree bound of projective subschemes

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give explicit Betti tables for almost maximal cases. Interesting examples are provided to understand our main results.


Introduction
Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n and codimension e defined over an algebraically closed field of arbitrary characteristic with the ideal sheaf I X . Let R X = k[x 0 , . . . , x n+e ]/I X be the homogeneous coordinate ring of X where I X = ⊕ ∞ m=0 H 0 (P n+e , I X (m)) is the saturated homogeneous ideal. Among the numerical invariants of X are the degree deg(X), the Castelnuovo-Mumford regularity reg(R X ) and the reduction number r(X) (which is defined as the reduction number of the homogeneous coordinate ring R X ). The complexity of R X is reflected in those invariants. The Castelnuovo-Mumford regularity is the height of the Betti table of R X and the reduction number r(X) is the least number among the maximal degrees of minimal generators of R X as a module over its Noether normalizations.
There have been several results on the relations between these invariants. For examples, we always have (see [21]) (1) 1 ≤ r(X) ≤ reg(R X ).
We can generalize the inequality (1) as follows: If R X is d-regular until the e-th step in the minimal free resolution, then It is interesting to mention that in many cases, the reduction number is smaller than d as shown by example 3.6(Ulrich's example), example 5.6 and example 5.7.
On the other hand, the degree upper bound can be read off from the Betti table. For example, the multiplicity conjecture( [17]), which was proved by Eisenbud and Schreyer, M. Boij and J. Soderberg ( [10], [6], [11]) gives us the upper bound with the maximal cases, which are arithmetically Cohen-Macaulay with pure resolution.
Along this line, the reduction number also provides an upper bound for the degree of a projective subscheme X ⊂ P n+e . Actually, in this paper we prove that (3) deg(X) ≤ e + r(X) r(X) .
Furthermore we show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with (r(X) + 1)-linear resolution (see Theorem 3.1). This result can be thought as a generalization of the previous results due to Ahn and Kwak [2] where they proved for an algebraic set X ⊂ P n+e , if R X is d-regular until the e-th step, then deg(X) ≤ e+d d , and in case d = 2 and ch(k) = 0, they showed that deg(X) = e+2 2 if and only if X is arithmetically Cohen-Macaulay with 3-linear resolution. The 'almost maximal' cases are also investigated in this direction. We have the refined inequalities (4) deg(X) ≤ µ S (R X ) ≤ e + r(X) r(X) , where S ֒→ R X is a Noether normalization which determines the reduction number r(X) and µ S (R X ) is the number of minimal generators of R X as an S-module. The almost maximal cases with deg(X) = e+r(X) r(X) − 1 consist of arithmetically Cohen-Macaulay subschemes with µ S (R X ) = e+r(X) r(X) − 1 and non-arithmetically Cohen-Macaulay subschemes with µ S (R X ) = e+r(X) r(X) . We characterize both cases by the description of the initial ideal of I X with respect to the degree reverse lexicographic order and the syzygy of the Noether normalization S ֒→ R X (see Theorem 4.1 and Corollary 5.2).
Furthermore, if X is a reduced, irreducible projective variety with deg(X) = e+r(X) r(X) − 1, then we show that depth(R X ) ≥ dim(X) (Theorem 5.1) and there are only two types according to depth(R X ). In this direction, the possible explicit Betti tables of those varieties are obtained from that of the initial ideal of I X and applying the graded mapping cone sequence or cancellation principle (Proposition 4.6, Theorem 5.4). We also provide interesting examples with all possible Betti tables to illustrate our main results. The Betti tables are computed by using Macaulay 2 [13].

Preliminaries
2.1. Reduction number. Throughout this paper, k is an infinite field. Let S i = k[x i , . . . , x n+e ] be the polynomial ring over k, for i = 0, 1, . . . , n + e. Let I ⊂ S 0 be a homogeneous ideal of codimension e and put R = S 0 /I. The irrelevant homogeneous ideal of R is denoted by R + . Let J be a minimal homogeneous reduction of R + . The reduction number of R + with respect to J is r J (R + ) = min{t > 0 : R t+1 + = JR t + } = min{t > 0 : R t+1 = J t+1 }. This number is a measure for the complexity of the algebra R. It has deep relations with other invariants of the same type such as the Castelnuovo-Mumford regularity and the a-invariant. The latter is defined by a n+1 (R) = max{t : H n+1 R+ (R) t = 0} + 1. N.V. Trung proved that Proposition 2.1. [21, Proposition 3.2] a n+1 + n ≤ r J (R + ) ≤ reg(R).
On the other hand, the ideal J is minimally generated by n + 1 linear forms and after a change of variables, we may assume that J = (x e , . . . , x n+e ). Let S = S e = k[x e , . . . , x n+e ]. The natural homomorphism S → R is in fact an inclusion and is a Noether normalization of R, i.e., S is a polynomial k-algebra and R is a finitely generated S-module. The reduction number can be computed by the maximal degree of minimal generators of R over S. Namely, The reduction number of R is the least number among r J (R + ) for all minimal reduction J of R + and will be denoted by r(R).
Sometimes we also call r J (R + ) the reduction number of the Noether normalization S ֒→ R and denote it by r S (R). The finite S-algebra structure on R is particularly interesting and should provide an effective way to understand the structure of the algebra. We have some very first properties of minimal sets of generators of R over S.
Proof. As an S-module, R is generated by all monomials in x 0 , . . . , x e−1 of degree at most r S (R). There are e+rS (R)

rS (R)
such monomials and the inequality follows.
We can describe precisely a minimal set of monomials generating R. To do this, we fix the degree reverse lexicographic order on the monomial of S 0 . Then R is minimally generated over S by monomials which formulate a basis of the k-vector space S 0 /I + (x e , . . . , x n+e ). These are exactly the standard monomials with respect to the ideal I X + (x e , . . . , x n+e ). Recall that a monomial in S 0 is a standard monomial with respect to a homogeneous ideal J if it is not contained in the initial ideal of J. To sum up, we have . , x n+e ] is a Noether normalization of R through the natural map S ֒→ R. Then R is minimally generated as a module over S by the set {m ∈ R : m is a monomial in x 0 , . . . , x e−1 which is standard with respect to I X + (x e , . . . , x n+e )}.
Related to this lemma, the following Cohen-Macaulay criterion for the quotient R = S 0 /I in term of the initial ideal in(I) due to Bermejo-Gimenez is very useful in our proof in this paper.
The Castelnuovo-Mumford regularity together with the projective dimension are the height and the width of the Betti table.
We say that R has a d-linear resolution if β S0 ij (R) = 0 unless i = j = 0 or j = d − 1 and i ≥ 1. Obviously, if R has a d-linear resolution then I is generated by a set of forms of degree d.
Closely related to the regularity is the N d,p -property defined by Eisenbud-Goto [8] and Eisenbud-Green-Hulek-Popescu [9]. We say that M satisfies the N d,p -property (d ≥ 2) if β S0 i,j (M ) = 0 for all i ≤ p and j ≥ d. In other words, M satisfies that N d,p property if M is (d − 1)-regular up to degree p. Clearly M satisfies the N d,p -property for all d > reg(M ) and all p ≥ 0.
In the study of structure of modules with N d,p -property, a mapping cone sequence of homology groups has been used effectively. Recall the notation S i = k[X i , . . . , X n+e ], i = 1, 2, . . . , n + e. Through the inclusion S 1 ⊂ S 0 , any graded S 0 -module M is also a graded S 1 -module. The Koszul complex K(x 1 , . . . , x n+e ; M ) fits in a short exact sequence of complexes where K(x 0 , M ) is the Koszul complex of M with respect to the single element x 0 . Then there is a long exact sequence of Koszul homology This leads to a so-called graded mapping cone sequence obtained by Ahn-Kwak.
Theorem 2.6. [1, Theorem 3.2] Let M be a graded S 0 -module. There is a long exact sequence i−1 (M, k) i+j → · · · whose connecting homomorphisms δ's are induced from the multiplication by x 0 .
The following consequence follows immediately from the exact sequence in the theorem. In order to prove (d), we note that if β S0 ij (M ) = 0 then from the exact sequence (5) in Theorem 2.6, As an application of Corollary 2.7, we obtain an upper bound for the reduction number which is somehow stronger than Trung's upper bound in Proposition 2.1. Proof. It should be mentioned that R always satisfies the N d,e -property for some d ≤ reg(R). So this corollary gives a stronger upper bound than Trung's. We may assume that S e = k[x e , . . . , x n+e ] is a Noether normalization of R. By Corollary 2.7(d), R satisfies the N d,0 -property as an S e -module. In other words, β Se 0,j (R) = 0 for all j ≥ d and hence r(R) < d (see Lemma 2.2).

Notations and Conventions
Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n with the ideal sheaf I X . Let ) be the saturated homogeneous ideal and R X = k[x 0 , . . . , x n+e ]/I X be the homogeneous coordinate ring.
• The reduction number of X is the same as the reduction number of R X , i.e., r(X) = r(R X ).
• The (i, j)-th Betti number of X is • The Castelnuovo-Mumford regularity of X is reg(X) = reg(I X ) = reg(R X ) + 1.
• We say that X satisfies the N d,p -property if so does R X . In other words, X satisfies the N d,p property if R X is (d − 1)-regular up to the degree p. • A projective variety is always assumed to be an irreducible and reduced projective subscheme. Let X ⊂ P n+e be a projective variety of codimension e. We say that X has the minimal degree if deg(X) = e + 1. The variety X is a del Pezzo variety if it is arithmetically Cohen-Macaulay with deg(X) = e + 2 (almost minimal degree).
In the whole paper, we only consider the degree reverse lexicographic order on the monomials. The initial ideal of a homogeneous ideal I with respect to this order is denoted by in(I).

Degree upper bound in terms of reduction number and the maximal cases
The nature of the reduction number is a bound for the complexity of an algebra or the associated scheme as we can see in Proposition 2.1 and especially Lemma 2.2. In this section, we give more evidence for this observation by providing an upper bound for the degree in terms of the reduction number. The main result of this section is the following theorem. Proof. Let I X ⊂ S 0 be the saturated homogeneous defining ideal of X and R = S 0 /I X . Changing the variables if necessary, we may assume that Q = (x e , . . . , x n+e ) is a minimal reduction of the irrelevant ideal R + with the reduction number r Q (R) = r. In particular, S = S e = k[x e , . . . , x n+e ] ֒→ R is a Noether normalization of R. Let µ S (R) = dim k (R/S + R) be the minimal number of generators of R over S. Then we have where mult Q (R) is the multiplicity of R with respect to Q. The last inequality is due to Proposition 2.3. This proves the inequality. Now, if X is arithmetically Cohen-Macaulay with an (r + 1)-linear resolution then Eisenbud-Goto [8, Conversely, assume that deg(X) = e+r r . Then all inequalities in (5) are actually equalities, i.e., is the multiplicity of A with respect to QA. Then dim k (A/QA) = e QA (A) and A is Cohen-Macaulay (see [19,Theorem 17.11]). So R is Cohen-Macaulay. It remains to show that X has an (r + 1)-linear resolution. Since R is Cohen-Macaulay, we have by Proposition 2.1 that reg(R) = a n+1 (R) + n = r S (R) = r.
Hence reg(I X ) = reg(R) + 1 = r + 1. On the other hand, combining the fact µ S (R) = e+r r with Lemma 2.4, we can see that R is minimally generated over S by all the monomials x α0 0 . . . x αe−1 e−1 with α 0 + . . . + α e−1 ≤ r. They are all standard monomials with respect to I X + (x e , . . . , x n+e ) and the degree reverse lexicographic order on the monomials. Note also that, due to Bermejo-Gimenez (see Proposition 2.5), in(I X ) has a set of generators consisting of monomials in x 0 , . . . , x e−1 . Therefore . In particular, the minimal degree of generators of I X is r + 1. Combining this with the fact that reg(I X ) ≤ r + 1, we get that I X is generated by a set of forms of degree r + 1, hence R has an (r + 1)linear minimal free resolution.
The second part of Theorem 3.1 has been proven by Ahn-Kwak [2] under the assumption that X is an algebraic set satisfying the N 3,e -property and k is of characteristic zero. Later it is generalized for algebraic sets with the N d,e -property by Ahn-Han-Kwak [3] over fields of arbitrary characteristics. In the present paper, in place of N d,e -property we use the reduction number to get a better degree upper bound and a short proof for the characterization of closed subschemes of maximal degree.
If a projective closed subscheme satisfies the equivalent conditions in Theorem 3.1, i.e., deg(X) = e+r r , we say that it has a maximal degree. Combining Theorem 3.1 and its proof with the description of the Betti table of Cohen-Macaulay graded k-algebra with linear resolution (see [8,Proposition 1.7]), we obtain the following consequence.
Corollary 3.2. Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n, codimension e and reduction number r. Let I X ⊂ S 0 be the defining ideal of X and R X = S 0 /I X . Assume that (a) X has a maximal degree if and only if the initial ideal with respect to the degree reverse lexicographic order in(I X ) is generated by the set of all monomials in x 0 , x 1 , . . . , x e−1 of degree r + 1; (b) Suppose X has a maximal degree. The graded Betti numbers of X are otherwise.
(Zero dimensional schemes with maximal degree) Let Γ ⊂ P e be a nondegenerate finite scheme with the reduction number r. Since Γ is arithmetically Cohen-Maculay, the reduction number r is equal to reg R Γ . Therefore, it is r-normal and deg(Γ) ≤ e+r r (or using Theorem 3.1). In particular, deg(Γ) = e+r r if and only if there is no hypersurface of degree r containing Γ if and only if I Γ has a (r + 1) linear resolution.
For example, consider two distinct sets of 6 points in P 2 . one is that Γ 1 is not contained in a conic curve. In this case, r(Γ 1 ) = 2 and for a Noether normalization S → R Γ1 defining the reduction number The other is that Γ 2 is contained in a unique conic curve. In this case, r(Γ 2 ) = 3 and for a Noether normalization S → R Γ2 defining the reduction number r(Γ 2 ), we also have On the other hand, the proof of Theorem 3.1 and Corollary 3.2 induce the following combinatorial identity which is presented here for later usage.
x n+e ] be a monomial ideal generated by all monomials in x 0 , . . . , x e−1 of degree r +1. Then R = S 0 /I is a Cohen-Macaulay k-algebra with an (r +1)-linear minimal free resolution and deg(R) = e+r r (see Corollary 3.2). Moreover, (x e , . . . , x n+e ) is a minimal reduction of R + with the reduction number r. In particular, S = S e = k[x e , . . . , x n+e ] ⊂ R is a Noether normalization of R.
As an S-module, the Betti numbers of R are On the other hand, the Betti numbers of the S 0 -module R are Following Corollary 2.7, the Betti numbers of R over S and over S 0 are related by the equality for m ∈ Z. From the Betti table of R as an S-module above, we have In general, the reduction number is smaller than the regularity index d in the N d,e -property. The following example shows that the reduction number is 1 and the regularity index can be arbitrarily large.
the reduction number r(R) = 1 and its minimal resolution is ...as an S-module.
On the other hand, the Betti table of R as an S 0 -module is given in the following. So reg(R) = t, where the symbol − means that the corresponding Betti number is zero.
In the second part of this section, we will apply Theorem 3.1 and its consequences to study projective varieties with small reduction number. Recall that a projective variety X ⊂ P n+e of codimension e has a minimal degree if deg(X) = e + 1. Proof. Let X be a variety with reduction number r(X) = 1. The degree of X is bounded above by e + 1 due to Proposition 2.3. Hence deg(X) = e + 1 and X has a minimal degree. Furthermore, reg(X) = 2 by [8]. This shows (b) ⇒ (a) and (a) ⇒ (c).
On the other hand, if reg(X) = 2 then reg(R) = 1, where R is the homogeneous coordinate ring of X. So the implication (c) ⇒ (b) follows immediately from Proposition 2.1.
The case of projective closed subschemes of reduction number 2 is much more complicated. The del Pezzo varieties are interesting examples of such subschemes. Recall that a projective variety X ⊂ P n+e is a del Pezzo variety if it is arithmetically Cohen-Macaulay with deg(X) = codim(X) + 2 (i.e. almost minimal degree).
Example 3.8. Let X ⊂ P n+e be a del Pezzo variety of dimension n. The Betti table of the homogeneous coordinate ring of X is [18] and also the next proposition). So X satisfies the N 3,e -property but not the N 2,e -property. The reduction number of X is at most 2 by Corollary 2.8. On the other hand, as a del Pezzo variety does not have a minimal degree, its reduction number is at least 2 by Corollary 3.7. This concludes that r(X) = 2.
The next proposition provides information on Betti table of any arithmetically Cohen-Macaulay subscheme of reduction number 2. Proposition 3.9. Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n. Assume that X is arithmetically Cohen-Macaulay with reduction number r(X) = 2. Then the Betti table of the homogeneous coordinate ring of X (over S 0 ) is In particular, deg(X) = e + 1 + β e,2 .
Proof. Let R be the homogeneous coordinate ring of X. By changing the variables, we might assume that S = k[x e , . . . , x n+e ] is a Noether normalization of R and reduction number r S (R) = r(R) = 2.

The arithmetically Cohen-Macaulay varieties with almost maximal degree
Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n and the reduction number r. The degree of X is always bounded above by e+r r and we have seen in Theorem 3.1 a characterization for those subschemes whose degree attains the maximal value. In the next two sections we will investigate the almost maximal cases, namely, when We keep the notations S i , I X , R X = S 0 /I X as in the previous section. Changing the variables if necessary, we assume that J = (x e , . . . , x n+e ) is a minimal reduction of (R X ) + with reduction number r J ((R X ) + ) = r(R X ) = r. Then S = S e = k[x e , . . . , x n+e ] is a Noether normalization of R X . Moreover, deg(X) = mult J (R X ), the latter is the multiplicity of R X with respect to the ideal J. Combining this observation with Proposition 2.3, there are bounds for the minimal number of generators of R X as an S-module (7) deg(X) ≤ µ S (R X ) ≤ e + r r .
If deg(X) = e+r r − 1 then there are only two possibilities for µ S (R X ), namely, 1) µ S (R X ) = e+r r − 1: the arithmetically Cohen-Macaulay case; 2) µ S (R X ) = e+r r : the non-arithmetically Cohen-Macaulay case. This section is devoted to study the arithmetically Cohen-Macaulay case, while the latter will be considered in the next section.
An arithmetically Cohen-Macaulay subscheme of almost maximal degree is characterized in the following theorem.
The fact dim k (I X ) r = 1 implies that the vector space of degree r component of the initial ideal of I X has dimension one too. Hence µ S (R X ) ≥ e+r r − 1 due to Lemma 2.4. If deg(X) = µ S (R X ) = e+r r then I X has an (r + 1)-linear resolution by Theorem 3.1. This is impossible because (I X ) r = 0. Therefore (b) ⇒ (c): Suppose that R X is Cohen-Macaulay. Due to Bermejo-Gimenez (see Proposition 2.5), the initial ideal in(I X ) is minimally generated by a set B of monomials in x 0 , . . . , x e−1 . Moreover, the generating set B contains a monomial m of degree r since dim k (I X ) r = 1.
On the other hand, as we have seen in the proof of (b) ⇒ (a), the S-module R X is generated by all monomials in x 0 , . . . , x e−1 of degrees at most r except the monomial m. As these are all the standard monomials with respect to in(I X ) + (x e , . . . , x d ), then B must contain all monomials in x 0 , . . . , x e−1 of degree r + 1 which are not multiples of m. This proves (c).   The defining ideal of X is generated by a quadric and two cubics. The S 0 -minimal free resolution of the homogeneous coordinate ring R X is Betti table   0  1  2  0 1 Assume that S 2 = k[x 2 , x 3 , x 4 ] is a Noether normalization of R. Then as an S 2 -module, In particular, X has an almost maximal degree 2+2 2 − 1 and r(X) = 2. Let C be a generic hyperplane section of X ⊂ P 4 which is a smooth curve of degree 5 and genus 2 in P 3 . Then, C is also arithmetically Cohen-Macaulay with r(C) = 2 (see [15,Example 6.4.2]). The Betti table of its homogeneous coordinate ring R C is the same as that of R X .
If we do not require the subschemes to be reduced and irreducible, examples are found easily. The following example comes from Theorem 4.1.
has the regularity r and I has the regularity r + 1, the regularity of K/uJ is r + 1. This together with the fact K/uJ being generated by degree (r + 1)-elements imply that K/uJ has an (r + 1)linear minimal free resolution. Now we show that the homomorphism Tor S0 i+1 (k, K/uJ) i+j → Tor S0 i (k, S 0 ) i+j−r in the long exact sequence above is actually zero. Indeed, if j = r + 2 then Tor S0 i+1 (k, K/uJ) i+j = 0. If j = r + 2 then Tor S0 i (k, S 0 ) i+j−r = 0. This proves the claim. Consequently, we have a short exact sequence 0 → Tor S0 i (k, S 0 ) i+j−r → Tor S0 i (k, I) i+j → Tor S0 i (k, K/uJ) i+j → 0, for all i, j. This particularly implies that β S0 ij (I) = β S0 i,j−r (S 0 ) + β S0 ij (K/uJ). In order to compute the Betti numbers of K/uJ, we use the short exact sequence Again, there is a long exact sequence . . Note that all J[−r], K and K/uJ have (r + 1)-linear resolutions. By analogous argument as in the first part of the proof, we obtain a short exact sequence . Now, using Corollary 3.2, finally we get an explicit formula for the Betti numbers of I, namely, otherwise. Lemma 4.5 does not only give the explicit Betti table for a particular subscheme of almost maximal degree but is also very useful when we compute the Betti table for the general case. We will see this in the following proposition.
Proposition 4.6. Let X ⊂ P n+e be a non-degenerate closed subscheme of dimension n and reduction number r. Assume that X is arithmetically Cohen-Macaulay with deg(X) = e+r r − 1. The Betti table of the homogeneous coordinate ring of X is (in the following table, we write only rows with some possibly non-zero entries) Proof. Let R = S 0 /I X be the homogeneous coordinate ring of X. We assume that S = k[x e , . . . , x n+e ] ⊂ R is a Noether normalization of R with the reduction number r S (R) = r. We consider R as an S 0 -module and as an S-module.
As an S-module, the Betti number of R is otherwise.
Using Corollary 2.7(a) which relates the Betti numbers of R over S with those over S 0 , we are able to find a nice relation between Betti numbers of R over S 0 , namely, On the other hand, by Theorem 4.1, the initial ideal in(I X ) with respect to the degree reverse lexicographic order is generated by all monomials in x 0 , x 1 , . . . , x e−1 of degree r + 1 and a monomial in x 0 , x 1 , . . . , x e−1 of degree r. Lemma 4.5 applies to in(I X ) and we have β S0 ij (in(I X )) = 0, for either j = r, r + 1, or j = r, i > 0, or j = r + 1, i ≥ e. Now, by comparing the Betti numbers of I X and its initial ideal (see, for example, [14, Corollary 1.21]), we get for either j = r, r + 1, or j = r, i > 0, or j = r + 1, i ≥ e. This implies that Combining these with Lemma 3.5, we therefore obtain

The non-arithmetically Cohen-Macaulay varieties with almost maximal degree
In this section we consider projective subschemes with almost maximal degree which are not arithmetically Cohen-Macaulay. It is much more complicated to explore this class of subschemes than the arithmetically Cohen-Macaulay subschemes in the previous section. Fortunately, in the case of reduced and irreducible projective subschemes, we have the following key theorem. Proof. Let I ⊂ S 0 = k[x 0 , . . . , x n+e ] be the defining prime ideal of X and R = S 0 /I. Assume that S = S e = k[x e , . . . , x n+e ] is a Noether normalization of R such that the reduction number r S (R) = r. It is enough to prove the assertion for the case X being not arithmetically Cohen-Macaulay.
The proof consists of several steps. Note that we fix the degree reverse lexicographic order on the monomials of S 0 .
Step 1. We show that the initial ideal in(I) has a minimal set of generators consisting of all monomials in Since X is not arithmetically Cohen-Macaulay, we have µ S (R) = e+r r . It together with Proposition 2.3 and Lemma 2.4 shows that the monomials in T 0 ∪ T 1 ∪ . . . ∪ T r are all monomials not contained in in(I) + (x e , . . . , x n+e ). This also shows that in(I) has a minimal set of generators consisting of T r+1 and some monomials u 1 v 1 , . . . , u s v s where u i ∈ T j for some j ≤ r and v i ∈ S + . Here it is worth noting that deg(u i ) > 0 since dim(S 0 /I) = dim(S 0 / in(I)).
So there is only one monomial of degree at most r in (u 1 , u 2 , . . . , u s , T r+1 ) which is not in in(I). This implies that u 1 = u 2 = . . . = u s which is of degree r. Denote it by u.
Step 2. The aim of this step is to prove that the equivalence classes in R of the monomials in (T 0 ∪ T 1 ∪ . . . ∪ T r ) \ {u} are linearly independent over S. Let f = f 1 u 1 + . . . + f N u N ∈ I. We can assume in addition that f 1 , . . . , f N are homogeneous polynomials such that f is also homogeneous. Obviously the monomials u 1 , . . . , u N ∈ S 0 are linearly independent over S, so f = 0. Let in(f ) = λm 1 m 2 , where λ ∈ k, λ = 0, and m 1 ∈ (T 0 ∪ T 1 ∪ . . . ∪ T r ) \ {u}, m 2 ∈ S. This contradicts to the fact that in(f ) lies in in(I) which is minimally generated over S by T r+1 ∪ {uv 1 , . . . , uv s }. This completes the proof for Step 2.
Step 3. We prove that in(I) is minimally generated by T r+1 ∪ {uv 1 } using the fact that I is a prime ideal.
As we have shown in Step 1, the initial ideal in(I) is minimally generated by the monomials in T r+1 and some monomials uv 1 , . . . , uv s , where u ∈ T r and v 1 , . . . , v s ∈ S + . For the convenience, we denote the monomials in T r+1 by m 1 , . . . , m t , where t = e+r r+1 . Let h 1 , . . . , h s be part of the reduced Gröbner basis with in(h j ) = uv j . Observe that the polynomials h 1 , . . . , h s are irreducible as I is a prime ideal.
Since no trailing terms of any polynomial in the Gröbner basis lie in the initial ideal in(I), we write where q i , q i1 , . . . , q iN are homogeneous polynomials in S and in(q i ) = v i . The irreducibility of h i implies particularly that q i , q i1 , . . . , q iN have no common factors of positive degree. Assume that s ≥ 2. Then which is in I. By Step 2, the equivalence classes of the monomials u 1 , . . . , u N in R are linearly independent over S. Hence q 2 q 1j −q 1 q 2j = 0 for all j = 1, . . . , N . Since q i , q ij ∈ S have unique irreducible factorizations and q i , q i1 , . . . , q iN have no common factors of positive degree for each i = 1, 2, it implies that q 1 = λq 2 for some λ ∈ k. This is impossible because it implies v 1 = in(q 1 ) = λ in(q 2 ) = λv 2 . Therefore s = 1 and in(I) is minimally generated by T r+1 ∪ {uv 1 }.
Step 4. Finally, we show that depth(R) = n. By Step 3, the initial ideal in(I) is minimally generated by T r+1 ∪ {uv 1 }. We have a short exact is Cohen-Macaulay of dimension n + 1, we obtain depth(S 0 / in(I)) ≥ n.
In Theorem 5.1, we really use the fact that X is a projective variety, i.e., reduced and irreducible closed subscheme. We will see latter in Example 5.9 that this assumption is necessary and can not be omitted.
The following consequence follows from the proof of Theorem 5.1.
(e) R, as an S-module, has the graded Betti numbers On the other hand, β S 0,j (R) is bounded above by the number of monomials in x 0 , . . . , x e−1 of degree j, i.e., This implies that for all j = 0, 1, . . . , r. Moreover, we know by Theorem 5.1 that R has depth n. The Auslander-Buchsbaum formula then gives us proj.dim S0 (R) = depth(S 0 ) − depth(R) = e + 1. Hence proj.dim S (R) = 1 by Corollary 2.7(c). Equivalently, β S i,j (R) = 0 for all i > 1, j ≥ 0, and there is a positive integer d > 0 such that The minimal free S-resolution of R thus is It remains to show that d = reg(R), or equivalently, d ≥ r. In the proof of Theorem 5.1, we have seen that the initial ideal with respect to the degree reverse lexicographic order in(I) is minimally generated by all monomials in x 0 , . . . , x e−1 of degree r + 1 and a monomial uv where v ∈ S + and u is a monomial in x 0 , . . . , x e−1 of degree r (see Step 3 in the proof of Theorem 5.1). Moreover, the equivalence classes in R of the monomials in x 0 , . . . , x e−1 of degree at most r except u are linearly independent over the ring S. Therefore, d + 1 ≥ deg(uv) ≥ r + 1, or d ≥ r.
So over the Noether normalization S = S e of the homogeneous coordinate ring R, the Betti table of a projective variety X with almost maximal degree is described precisely. It depends on whether the variety is arithmetically Cohen-Macaulay (ACM) or non-arithmetically Cohen-Macaulay (non-ACM) and is either . It worth noting that for any t > 0, the ideal J t defines a subscheme of maximal degree. Now using Corollary 3.2, we easily complete the proof.
Proof. Let I X ⊂ S 0 be the saturated homogeneous defining ideal of X and R = S 0 /I X . Changing the variables if necessary, we may assume that Q = (x e , . . . , x n+e ) is a minimal reduction of the irrelevant ideal R + with the reduction number r Q (R) = r. In particular, S = S e = k[x e , . . . , x n+e ] ֒→ R is a Noether normalization of R. The algebra R is particularly a finitely generated module over S e−i for i = 0, 1, . . . , e. By Corollary 5.2, the initial ideal with respect to the degree reverse lexicographic order in(I) is generated by all monomials in x 0 , . . . , x e−1 of degree r + 1 and a monomial uv, where u is a monomial in By the Cancellation Principle (see [14,Corollary 1.21] or [16,Section 3.3]), there is for each m ≥ 0 a complex of k-vector spaces ), for any i, j. Now using Lemma 5.3, we obtain the vanishing of Betti numbers, namely, β S0 ij (I X ) = 0, for either j = r + 1, reg(R) + 1, or j = r + 1, i ≥ e, or j = reg(R) + 1, i > e. Moreover, it also shows that in the complex V •,m above, almost all vector spaces are zero except V i,r+1+i and V i,reg(R)+1+i .
To ease the presentation, we consider two cases according to the difference between the regularity and the reduction number.
Case 1: reg(R) is either r or r + 1.
This is an application of the Cancellation Principle. Using the notations in the first part of the proof, we have V i,m = 0 for all m, i ≥ 0 such that m − i = r + 1, reg(R) + 1. In particular, V i,r+2+i = 0 and V i,reg(R)+i = 0. Consequently, we obtain H i (V •,m ) ≃ V i,m for all i, m, which induces by the Cancellation Principle the equality β S0 ij (I X ) = β S0 ij (in(I X )). Therefore the Betti number of the homogeneous coordinate ring of X can be computed using Lemma 5.3, namely, Example 5.5. Let C 1 be a smooth elliptic curve in P 3 of degree 5. Then, C 1 is an isomorphic projection from a complete embedding of an elliptic curve C 0 ֒→ P 4 . So, C 1 is m-normal for all m ≥ 2, but not a linearly normal. Thus, we have β 1,1 (C 1 ) = h 0 (I C1 (2)) = 0, reg(C 1 ) = 3, the reduction number r(C 1 ) = 2 and deg(C 1 ) = e+r r − 1 = 5. The Betti table of its homogeneous coordinate ring is   0  1  2  3  0  1  ---1  ----2  -5  5  1 This Betti table corresponds to (a) in Theorem 5.4.
Example 5.8. Let C 3 be the rational curve in P 4 defined as the image of the map P 1 → P 4 (s, t) → (s 9 , s 8 t − st 8 , s 7 t 2 − s 2 t 7 , s 6 t 3 + s 5 t 4 + s 4 t 5 + s 3 t 6 + s 8 t + s 7 t 2 , t 9 ) Using Macaulay 2, we find that the Betti table of C 3 is Let S 0 = k[x 0 , x 1 , x 2 , x 3 , x 4 ] and I be the defining ideal of C 3 . The homogeneous coordinate ring R = S 0 /I has a Noether normalization k[x 0 , x 4 ] → R and by computation, R/(x 0 , x 4 ) ≃ k[x 1 , x 2 , x 3 ]/(x 1 , x 2 , x 3 ) 3 . Hence the reduction number of C 3 is r(C 3 ) = 2. Since deg(C 3 ) = 9, the curve C 3 has an almost maximal degree. Moreover, C 3 is not arithmetically Cohen-Macaulay, and reg(C 3 ) = 6 and the Betti table of C 3 corresponds to (c) in Theorem 5.4.
We have proved in Theorem 5.1 that if a non-degenerate projective variety has almost maximal degree then its coordinate ring has depth at least the dimension of the variety. In the next example, we will see that projective varieties can not be replaced by projective subschemes in this theorem. The idea of the example comes from the proof of Theorem 5.1.
Example 5.9. Let S 0 = k[x 0 , . . . , x n+e ] as from beginning. Let r > 0 and let u be a monomial in x 0 , . . . , x e−1 of degree r. As in the proof of Theorem 5.1, we denote the set of all monomials in x 0 , . . . , x e−1 of degree r + 1 by T r+1 . Let I be the monomial ideal generated by T r+1 and ux e , . . . , ux n+e . Clearly I = (T r+1 , u) ∩ (T r+1 , x e , . . . , x n+e ).