Lifting the Determinantal Property

In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformation. We also study the transfer of the property of being standard or good determinantal under basic double links. Introduction Standard and good determinantal schemes are a large family of projective schemes, to which belong many varieties that have been classically studied. For example the Veronese variety, rational normal scrolls, rational normal curves, and some Segre varieties are good determinantal schemes. Standard determinantal schemes are cut out by the maximal minors of a matrix of forms (see Definition 1). In particular they are arithmetically Cohen-Macaulay, and their saturated ideal is resolved by the Eagon-Northcott complex. Good determinantal schemes are standard determinantal schemes that are locally a complete intersection outside a subscheme. Ideals of minors have been the object of extensive study in commutative algebra. These families were studied from the geometric viewpoint by Kreuzer, Migliore, Nagel, and Peterson in [12]. In this article, they introduced the definition of standard and good determinantal schemes that we use. The relevance of standard and good determinantal schemes in the context of liaison theory became clear in [9], where it was shown that standard determinantal schemes belong to the Gorenstein-liaison class of a complete intersection. In this note we study standard and good determinantal schemes and their general hyperplane sections. The property of being standard or good determinantal is preserved when taking a general hyperplane section. So we ask whether every arithmetically CohenMacaulay scheme whose general hyperplane section is good determinantal is itself good determinantal. The answer is negative. In Proposition 7, Example 9, and Proposition 15 we produce examples of schemes which are not standard determinantal, and whose general hyperplane section (or whose Artinian reduction) is good determinantal. In Proposition 10 we show that a section of the schemes of Proposition 7 by a number of generic hyperplanes is good determinantal up to flat deformation. Then we discuss how the property of being standard or good determinantal is preserved under basic double linkage. In Lemma 18 and I am grateful to J. Migliore and A. Conca for useful discussions. Part of the research in this paper was carried out while the author was a guest at the Max Planck Institut für Mathematik in Bonn. The computer algebra system CoCoA [3] was used for some of the computations. 1 2 ELISA GORLA Lemma 19 we prove that under some assumptions the property is preserved. In Proposition 21 and Proposition 22 we show that in other cases the property is not preserved under basic double linkage. In Example 24 we show how to combine the results about basic double links obtained so far. We produce a family of schemes via basic double link from the family of Proposition 15, and we prove that the schemes we produced are not standard determinantal, but their general hyperplane sections are good determinantal. Finally, we discuss the property of being standard or good determinantal in a flat family. This is motivated by the observation that we can study flat families all of whose elements are hyperplane section of a given scheme by a hyperplane that meets it properly. We show by means of examples that we can have a flat family which contains a non standard determinantal scheme and whose general element is standard determinantal, or the other way around. In Proposition 26 we give sufficient conditions on a section of a scheme S by a hyperplane that meets it properly that force a general hyperplane section of S to be good determinantal. We saw that a scheme S with good determinantal general hyperplane section does not need to be good determinantal. In Theorem 27 we show that S is good determinantal up to flat deformation. 1. Standard and good determinantal schemes Let S be a scheme in P = Pnk , where k is an algebraically closed field of characteristic zero. Let IS be the saturated homogeneous ideal corresponding to S in the polynomial ring R = k[x0, . . . , xn]. We denote by m the homogeneous irrelevant maximal ideal of R, m = (x0, . . . , xn). Let T be a scheme that contains S. We denote by IS|T the ideal of S restricted to T , i.e. the quotient IS/IT . We often write aCM for arithmetically Cohen-Macaulay. In this note we study schemes whose general hyperplane section is standard or good determinantal. The following definition was given in [12] for schemes, i.e. for saturated ideals. Here we extend it to include Artinian ideals. Definition 1. An ideal I ⊆ k[x0, . . . , xn] of height c is standard determinantal if it is generated by the maximal minors of a matrix M of polynomials of size t× (t+ c− 1), for some t ≥ 1. A standard determinantal scheme S ⊆ P of codimension c is a scheme whose saturated ideal IS is standard determinantal. A standard determinantal ideal I is good determinantal if after performing invertible row operations on the matrix M and then deleting a row, the ideal generated by the maximal minors of the (t − 1) × (t + c − 1) matrix obtained is standard determinantal (that is, it has height c+ 1). In particular, we formally include the possibility that t = 1, i.e. a complete intersection is good determinantal. A scheme S is good determinantal if its saturated ideal IS is good determinantal. Let S be a standard determinantal scheme with defining matrix M = (Fij). We assume without loss of generality that M contains no invertible entries. Let U = (uji) be the transposed of the matrix whose entries are the degrees of the entries of M . U is the LIFTING THE DETERMINANTAL PROPERTY 3 degree matrix of S. We adopt the convention that the entries of U increase from right to left and from top to bottom: uji ≥ ulk, if i ≤ k and j ≥ l. S can be regarded as the degeneracy locus of a morphism φ : t ⊕ i=1 R(bi) −→ t+c−1 ⊕ j=1 R(aj). Set a1 ≤ . . . ≤ at+c−1 and b1 ≤ . . . ≤ bt. Then φ is described by the transposed of the matrix M , and uji = aj − bi. In [12] the following result is proven. It gives two equivalent definition of good determinantal scheme that will be useful in the sequel. Theorem 2. Let S ⊆ P be a projective scheme. Let c ∈ Z, c ≥ 2. The following are equivalent: (1) S is good determinantal of codimension c, (2) S is the zero-locus of a regular section of the dual of a first Buchsbaum-Rim sheaf of rank c, (3) S is standard determinantal and locally a complete intersection outside a subscheme T ⊆ S of codimension c+ 1 in P. 2. Lifting the determinantal property In this note, we address the question of whether it is possible to lift the property of being standard or good determinantal from a general hyperplane section of a scheme to the scheme itself. For schemes of codimension 2, the Hilbert-Burch Theorem states that being standard determinantal is equivalent to being arithmetically Cohen-Macaulay. So this question is a natural generalization of the questions that we investigated in [6]. Before starting our discussion, we would like to observe that the good determinantal property does not behave as well as the standard determinantal property under hyperplane sections by a hyperplane that meets the scheme properly. In fact, any hyperplane section of a standard determinantal subscheme of P by a hyperplane that meets it properly is a standard determinantal subscheme of P. It is not true in general that every hyperplane section of a good determinantal subscheme of P by a hyperplane that meets it properly is a good determinantal subscheme of P. However, a general hyperplane section is good determinantal. Next, we see an example when this is the case. The example of the scheme Z ⊆ P supported on a point that is standard but not good determinantal is Example 4.1 in [9]. Example 3. Let C ⊆ P be a curve whose homogeneous saturated ideal is given by the maximal minors of ( x0 x1 + x4 0 x2 0 x1 x2 x0 + x1 ) . One can check that C is one-dimensional, hence standard determinantal. C is a cone over a zero-dimensional scheme supported on the points [0 : 0 : 0 : 1] and [0 : 1 : 0 : −1]. 4 ELISA GORLA The curve C is indeed good determinantal, since deleting a generalized row we obtain the matrix of size 1× 4 ( x0 x1 + αx4 x2 x0 + x1 + αx2 ) for a generic value of α. For α 6= 0 the entries form a regular sequence, since they are linearly independent linear forms. Therefore they define a complete intersection, that is a standard determinantal scheme, and C is good determinantal. Let H be a general linear form. In particular, we can assume that the coefficient of x3 in the equation of H is non-zero. Intersecting C with H we obtain a subscheme X of P, whose saturated homogeneous ideal IX is generated over k[x0, x1, x2, x4] by the maximal minors of ( x0 x1 + x4 0 x2 0 x1 x2 x0 + x1 ) . One can show that X is good determinantal following the same steps as for C. Indeed, C is just a cone over X. Let H = x4. Intersecting C with H we obtain a subscheme Z of P , whose saturated homogeneous ideal IZ is generated over k[x0, . . . , x3] by the maximal minors of ( x0 x1 0 x2 0 x1 x2 x0 + x1 ) . IZ = I 2 P for P = [0 : 0 : 0 : 1], hence Z is a zero-dimensional scheme supported on the point P . Then Z is standard determinantal and a section of C by a hyperplane that meets it properly. However, Z is not good determinantal. In fact, deleting a generalized row we obtain the matrix of size 1× 4 ( x0 x1 x2 x0 + x1 + αx2 ) whose entries generate the ideal (x0, x1, x2) of codimension 3 < 4. Every standard determinantal scheme is arithmetically Cohen-Macaulay. Moreover, the two families coincide for schemes of codimension 1 or 2, while for codimension 3 or higher the family of arithmetically Cohen-Macaulay schemes strictly contains the family of standard determinantal schemes. From the results in [8] one can easily obtain a sufficient condition for a scheme V ⊆ P to be arithmetically Cohen-Macaulay in terms of the graded Betti numbers of a general hyperplane section of V . If a general hyperplane section of V is standard determinantal, the condition can be expressed in terms of the entries of its degree matrix. Notice that since the graded Betti numbers of a hyperplane section of V are the same for a general choice of the hyperplane, the degree matrix is also the same for a general choice of the hyperplane. Corollary 4. Let V ⊆ P be a projective scheme. Assume that a general hyperplane section of V is a standard determinantal subscheme of P with degree matrix U = (uji)i=1,...,t; j=1,...,t+c−1. If either dimV ≥ 2 or u1,t + · · ·+ uc−1,t ≥ n+ 1 then V is arithmetically Cohen-Macaulay. LIFTING THE DETERMINANTAL PROPERTY 5 Proof. If dim(V ) ≥ 2 and a general hyperplane section of V is arithmetically CohenMacaulay, then V is arithmetically Cohen-Macaulay. We can then reduce to the case when V is one-dimensional. Let H be a general hyperplane, and let C = V ∩ H . From Theorem 3.16 of [8] it follows that the minimum degree of a minimal generator of IC that is not the image of a minimal generators of IV under the standard projection is b ≥ u1,1 + · · ·+ ut,t + ut+1,t + · · ·+ ut+c−1,t − n = = u1,t + · · ·+ uc−1,t + uc,1 + uc+1,2 + · · ·+ ut+c−1,t − n ≥ uc,1 + uc+1,2 + · · ·+ ut+c−1,t + 1. In particular, it is bigger than the maximum uc,1 + uc+1,2 + · · · + ut+c−1,t of the degrees of the minimal generators of IC . Then all the minimal generators of IC are images of the minimal generators of IV and V is arithmetically Cohen-Macaulay. As we mentioned, every arithmetically Cohen-Macaulay scheme of codimension 2 is standard determinantal. So Corollary 4 gives a sufficient condition to conclude that V is standard determinantal if codim(V ) = 2. Remark 5. Let V be a projective scheme. If dim(V ) ≥ 2 and a general hyperplane section of V is aCM, then V is aCM. Therefore the graded Betti numbers of V coincide with the graded Betti numbers of a general hyperplane section of V . Moreover, for a scheme of codimension 2 the property of being standard determinantal can be decided by checking the graded Betti numbers. Hence if dim(V ) ≥ 2 and codim(V ) = 2, we can decide whether V is standard determinantal by looking at the graded Betti numbers of a general hyperplane section. However, if codim(V ) ≥ 3 then the property of being standard determinantal cannot in general be decided by looking at the graded Betti numbers. In other words, there are schemes which are not standard determinantal, but have the same graded Betti numbers as a standard determinantal scheme (see e.g. Example 9). In very special cases the graded Betti numbers of a homogeneous ideal I can force the ideal to be standard determinantal, even when the codimension is 3 or higher. The next is an easy example of this phenomenon. Example 6. Let R = k[x1, . . . , xn], m = (x1, . . . , xn). Let I ⊆ R be a homogeneous ideal with graded Betti numbers · · · −→ R(−t)( n+t−1 t ) −→ I −→ 0. Then Ij = 0 for all j < t and dimIt = ( n+t−1 t ) = dim(m)t. Therefore I = m , so it is the ideal of maximal minors of the t× (t+ n− 1) matrix