By Elena Guardo, Adam Van Tuyl
This short provides an answer to the interpolation challenge for arithmetically Cohen-Macaulay (ACM) units of issues within the multiprojective area P^1 x P^1. It collects a few of the present threads within the literature in this subject with the purpose of delivering a self-contained, unified creation whereas additionally advancing a few new principles. The proper buildings concerning multiprojective areas are reviewed first, via the elemental 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 will be computed and, for this reason, resolve the interpolation challenge. In next chapters, they give thought to fats issues and double issues in P^1 x P^1 and exhibit tips on how to use their effects to respond to questions and difficulties of curiosity in commutative algebra. Throughout the booklet, chapters finish with a short ancient assessment, citations of comparable effects, and, the place correct, open questions which could motivate destiny learn. Graduate scholars and researchers operating in algebraic geometry and commutative algebra will locate this booklet to be a useful contribution to the literature.
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Extra info for Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1
Let X be a finite set of distinct points in P1 × P1 . 1 Arithmetically Cohen-Macaulay sets of points 43 where (i) n + 1 = m + p. p n m (ii) ∑pi=1 j2,i − ∑ni=1 j1,i + ∑m i=1 j0,i = ∑i=1 j2,i − ∑i=1 j1,i + ∑i=1 j 0, i = 0. p n (iii) |X| = − ∑m i=1 j0,i j0,i + ∑i=1 j1,i j1,i − ∑i=1 j2,i j2,i . 2 demonstrates, the depth of R/I(X) is either one or two. Both cases can occur, as we show in the next example. 5. A set consisting of a single point is ACM. To see this fact, we can make a change of coordinates, and then assume P = [1 : 0] × [1 : 0].
29. The entries in rows i = 2, . . , h − 1 are also a consequence of this theorem. Finally, note that for all (i, j) ≥ (h − 1, v − 1), α1∗ + · · · + αα∗1 = β1∗ + · · · + ββ∗1 = s. 31. 30 that all the values of the Hilbert function of X can be computed from αX and βX except for the values HX (i, j) with (1, 1) (i, j) (h − 2, v − 2). Thus, if either h ≤ 2 or v ≤ 2, then the entire Hilbert function can be computed from αX and βX . 32. 7 we have αX = (3, 2, 2) and βX = (2, 2, 2, 1). For these points αX∗ = (3, 3, 1) and βX∗ = (4, 3).
Hence, for this set of points, we know the following values of HX : ⎤ ⎡ 1 2 3 4 4 ··· ⎢2 ? 7 7 · · · ⎥ ⎥ ⎢ ⎥ ⎢ HX = ⎢3 6 7 7 7 · · ·⎥ . ⎢3 6 7 7 7 · · · ⎥ ⎦ ⎣ .. .. .. . . . . , namely, HX (1, 1) and HX (1, 2), cannot be determined from αX and βX . 33. It is possible for two sets of points to have the same αX and βX , but not the same Hilbert function. For example, let A1 , A2 , A3 be three distinct points of P1 , and let B1 , B2 , and B3 be another collection of three distinct points in P1 .
Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1 by Elena Guardo, Adam Van Tuyl