Box-shaped Matrices and the Defining Ideal of Certain Blowup Surfaces

In this paper, we generalize the notions of a matrix and its ideal of 2 × 2 minors to that of a box-shaped matrix and its ideal of 2 × 2 minors, and make use of these notions to study projective embeddings of certain blowup surfaces. We prove that the ideal of 2× 2 minors of a generic box-shaped matrix is a perfect prime ideal that gives the algebraic description for the Segre embedding of the product of several projective spaces. We use the notion of the ideal of 2× 2 minors of a box-shaped matrix to give an explicit description for the defining ideal of the blowup of P along a set of ( d+1 2 ) (d ∈ Z) points in generic position, embedded into projective spaces using very ample divisors which correspond to the linear systems of plane curves going through these points. 0. Introduction. Ideals of minors of a matrix have been thoroughly studied over many decades. They play a significant role in the study of projective varieties. It had been a major classical problem to show that the ideal of t× t minors of a generic matrix is a prime and perfect ideal. The proof for a general value of t was due to Eagon and Hochster from their important work in [16]. In the first part of this paper, we generalize the notions of a matrix and its ideal of 2×2 minors to that of a box-shaped-matrix and its ideal of 2×2 minors. Our main theorem in this section is the following theorem. Theorem 0.1 (Theorem 1.5). If A is a box-shaped matrix of indeterminates, then I2(A) is a prime ideal in k[A] (here, I2(A) is the ideal of 2× 2 minors of A). Coupled with previous work of Grone ([13]), we also show that the ideal of 2 × 2 minors of a generic box-shaped matrix is the defining ideal of a Segre embedding of the product of several projective space, namely P(V1) × . . . × P(Vn) →֒ P(V1 ⊗ . . . ⊗ Vn). This geometric 2000 Mathematics Subject Classification. Primary: 13C40, 14J26, Secondary: 14E25. Current address: Institute of Mathematics, P.O. Box 631 Bò Hô, Hà Nôi 10000, Vietnam. Email: [email protected]. 1 realization of the ideal of 2× 2 minors of a generic box-shaped matrix enables us to study its perfection (Theorem 1.8), its Hilbert function (Proposition 1.7), and gives a Gröbner basis (Theorem 1.11). Box-shaped matrices not only describe the Segre embedding of the product of several projective spaces, but also provide a new tool for the study of projective embeddings of certain blowup surfaces. This study is carried out in the second part of this paper. To be more precise, let X = {P1, . . ., Ps} be a set of s distinct points in P 2, and let IX = ⊕t≥αIt ⊆ R = k[w1, w2, w3] be the homogeneous decomposition of the defining ideal of X, and P2(X) the blowup of P2 centered at X. The second part of this paper studies the problem of finding systems of defining equations for P2(X) embedded in projective spaces by very ample divisors which correspond to the linear systems of plane curves going through the points in X. This problem has also been considered by several authors in the last ten years, such as [5], [6], [7], [10], [11], [18], [19] and [20]. A great deal of work has concentrated on an important special case, when s = ( d+1 2 ) for some positive integer d and the points in X are in generic position (cf. [5], [10], [11]). In this case, IX = Id ⊕ Id+1 ⊕ Id+2 ⊕ . . . is generated by Id (see [9]). We also address this situation. It is well known that, in our situation, all the linear systems It (for t ≥ d+1) are very ample (cf. [3], [7]), so they all give embeddings of P2(X) in projective spaces. If in addition, there are no d points of X lying on a line, then the linear system Id is also very ample. Under this assumption, Gimigliano studied the embedding of P2(X) given by the linear system Id, which results in a White surface ([10] and [11]). White surfaces had also been studied in the classical literature ([24] and [30]). Gimigliano showed that the defining ideal of a White surface is generated by the 3× 3 minors of a 3× d matrix of linear forms, and its defining ideal has the same Betti numbers as that of the ideal of 3 × 3 minors of a 3 × d matrix of indeterminates (which was given by the Eagon-Northcott complex). The embedding of P 2(X) given by the linear system Id+1 (which results in a Room surface) was then studied in detail by Geramita and Gimigliano ([5]). Geramita and Gimigliano were able to determine the resolution of the ideals defining the Room surfaces. They also proved that the defining 2 ideals of the embeddings of P2(X) given by the linear systems It are generated by quadrics, for all t ≥ d+ 1, but they were unable to write down those generators when t ≥ d+ 2. In [2] and [27], a method of finding a system of defining equations for a diagonal subalgebra from that of a bigraded algebra was given. This, together with results of [22] (which gives the equations for the Rees algebra of the ideal of a set of ( d+1 2 ) points in generic position), makes it possible, in theory, for one to write a system of defining equations for the embeddings of P2(X) given by the linear systems It for all t. However, this method has its disadvantages as pointed out in [14]. In the second part of this paper, we generalize Geramita and Gimigliano’s argument on the Room surfaces and give an explicit description of the defining ideals of the embeddings of P2(X) given by the linear systems It, for all t ≥ d+ 1. Our main result in this section is the following theorem. Theorem 0.2 (Theorem 2.6). Suppose t = d + n (n ≥ 1). Then the projective embedding of P2(X) given by the linear system It is generated by ( n+1 2 ) d linear forms and the 2 × 2 minors of a box-shaped matrix of linear forms. Throughout this paper, k will be our ground field. For simplicity, we assume that k is algebraically closed and of characteristic 0 (though many of our results are true for any field k). We also let P2 = P2 k be the projective plane over k. 1. Box-shaped matrices and their ideals of 2× 2 minors. The techniques we use in this section are inspired by those of [26] in his study of ideals of 2× 2 minors of a matrix. Box-shaped matrices Let S be a commutative ring that contains a field k. An n-dimensional array (n ≥ 2) A = (ai1...in)1≤ij≤rj , ∀j=1,...,n, can be realized as the box B = {(i1, . . . , in)|1 ≤ ij ≤ rj, ∀j}, 3 in which each integral point (i1, . . . , in) is assigned the value ai1...in . Definition. An n-dimensional array A, with its box-shaped realization B, is called an n-dimensional box-shaped matrix of size r1 × . . .× rn. We associate to each box-shaped matrix A of elements in S a ring SA = k[A], the subring of S obtained by adjoining the elements of A to the field k. Definition. Suppose A is an n-dimensional box-shaped matrix of size r1 × . . . × rn of elements in S. For each l = 1, 2, . . . , n, we call ai1...il...inaj1...jl...jn − ai1...il−1jlil+1...inaj1...jl−1iljl+1...jn ∈ SA, (where (i1, . . . , in) and (j1, . . . , jn) are any two integral points in B), a 2 × 2 minor about the l-th coordinate of A. A 2 × 2 minor of A is a 2 × 2 minor about at least one of its coordinates. We let I2(A) be the ideal of SA generated by all the 2 × 2 minors of A, and call it the ideal of 2× 2 minors of the box-shaped matrix A. From now on, unless stated otherwise, we focus our attention to box-shaped matrices of indeterminates. Suppose A = (xi1...in)(i1,...,in)∈B is an n-dimensional generic box-shaped matrix of size r1 × . . .× rn with its box-shaped realization B. For each l = 1, . . . , n, let Al = (xi1...in)(i1,...,in)∈B and il . Throughout this paper, to any box-shaped matrix A, we always associate box-shaped matrices Al, boxes Bl and all the ideals Il defined as above. The first crucial property of box-shaped matrices of indeterminates comes in the following lemma. Lemma 1.1. Suppose A = (xi1...in)(i1,...,in)∈B is a box-shaped matrix of indeterminates in S. Then, 4 (a) For any l 6= s ∈ {1, . . . , n}, we have Il ∩ Is = < I2(A), {xi1...in |(i1, . . . , in) ∈ Bl ∩Bs} > . (b) For any distinct elements l1, l2, . . . , lt of {1, 2, . . . , n} (2 ≤ t ≤ n), we have ∩j=1Ij = < I2(A), {xi1...in |(i1, . . . , in) ∈ ∩ t j=1Bj} > . Proof. (a) For convenience, we denote by LHS and RHS the left hand side and the right hand side of the presented equality, respectively. It is clear that RHS ⊆ LHS. We need to show the opposite direction. Let F ∈ LHS. Since F ∈ Il, we can write F = F ′+F ′′, where F ′ ∈ I2(Al), and F ′′ = ∑ (i1,...,in)∈Bl Fi1...inxi1...in . It suffices to show that F ′′ ∈ RHS. F ′′ certainly belongs to Is. Now, for (i1, . . . , in) ∈ Bl, we write Fi1...in in the form Fi1...in = ∑ (j1,...,jn)∈Bs Gi1...in,j1...jnxj1...jn +Gi1...in , where Gi1...in is independent of the indeterminates {xj1...jn |(j1, . . . , jn) ∈ Bs}. Then F ′′ = G+G′, where G = ∑ (i1,...,in)∈Bl Gi1...inxi1...in , and G = ∑ (i1,...,in)∈Bl,(j1,...,jn)∈Bs Gi1...in,j1...jnxi1...inxj1...jn = ∑ (i1, . . . , in) ∈ Bl, (j1, . . . , jn) ∈ Bs ( Gi1...in,j1...jnXi1...in,j1...jn + xi1...is−1jsis+1...inTi1...in,j1...jn )