By Elena Guardo, Adam Van Tuyl

This short offers an answer to the interpolation challenge for arithmetically Cohen-Macaulay (ACM) units of issues within the multiprojective house P^1 x P^1. It collects many of the present threads within the literature in this subject with the purpose of supplying a self-contained, unified advent whereas additionally advancing a few new rules. The proper structures on the topic of multiprojective areas are reviewed first, through the fundamental houses of issues in P^1 x P^1, the bigraded Hilbert functionality, and ACM units of issues. The authors then convey how, utilizing a combinatorial description of ACM issues in P^1 x P^1, the bigraded Hilbert functionality may be computed and, consequently, remedy the interpolation challenge. In next chapters, they give thought to fats issues and double issues in P^1 x P^1 and show easy methods to use their effects to respond to questions and difficulties of curiosity in commutative algebra. Throughout the publication, chapters finish with a quick ancient review, citations of similar effects, and, the place appropriate, open questions which may motivate destiny learn. Graduate scholars and researchers operating in algebraic geometry and commutative algebra will locate this ebook to be a priceless contribution to the literature.

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22 3 Points in P1 × P1 On P1 × P1 there exist two families of lines {HC } and {VC }, each parametrized by C ∈ P1 , with the property that if A = B ∈ P1 , then HA ∩ HB = 0/ and VA ∩ VB = 0, / and for all A, B ∈ P1 , HA ∩ VB = A × B is a point on P1 × P1 . We can thus view P1 × P1 as a grid with horizontal and vertical rulings. A point P = [a0 : a1 ] × [b0 : b1 ] ∈ P1 × P1 can be viewed as the intersection of the horizontal ruling defined by the degree (1, 0) line H = a1 x0 − a0 x1 and the vertical ruling defined by the degree (0, 1) line V = b1 y0 − b0 y1 .

Now the set Z still satisfies property ( ) and |π1 (Z)| < |π1 (X)|. By induction, Z is ACM, and hence proj-dim(I(Z)) = 1. Because the projective dimension of a free R-module is 0, proj-dim(R(−1, v)) = 0 and proj-dim(I(Z)(−1, 0) ⊕ R(0, −v)) = max{proj-dim(I(Z)), proj-dim(R(0, −v)} = proj-dim(I(Z)). 23 we have proj-dim(I(X)) ≤ 1. Because I(X) is not a free R-module, we thus have proj-dim(I(X)) = 1. 22). Thus, X is ACM as desired. 6, after a change of coordinates, we can assume that x0 and y0 form a regular sequence on R/I(X).

34. Let X be a set of distinct points in P1 × P1 with P ∈ X. A bihomogeneous form F ∈ R is a separator for P if F(P) = 0, but F(Q) = 0 for all Q ∈ X \ {P}. We call F a minimal separator for P if there does not exist a separator G for P with deg G ≺ deg F. If T ⊆ N2 is a subset, then let min T denote the set of minimal elements of T with respect to the partial order on N2 . 35. Let X be a set of distinct points in P1 × P1 . The degree of a point P ∈ X is the set degX (P) = min{deg F | F is a separator for P ∈ X}.