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Arithmetically Cohen-Macaulay Sets of Points in P^1 x P^1 by Elena Guardo, Adam Van Tuyl

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 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 rules.  The proper buildings relating to multiprojective areas are reviewed first, through the elemental homes of issues in P^1 x P^1, the bigraded Hilbert functionality, and ACM units of issues.  The authors then exhibit how, utilizing a combinatorial description of ACM issues in P^1 x P^1, the bigraded Hilbert functionality could be computed and, therefore, remedy the interpolation challenge.  In next chapters, they give thought to fats issues and double issues in P^1 x P^1 and show find out how to use their effects to reply to questions and difficulties of curiosity in commutative algebra.  Throughout the ebook, chapters finish with a quick ancient assessment, citations of similar effects, and, the place correct, open questions that could encourage destiny examine.  Graduate scholars and researchers operating in algebraic geometry and commutative algebra will locate this e-book to be a important contribution to the literature.

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H − 1 h h + 1 h + 2 · · · h + v − 3 h + v − 2 h + v − 2 · · ·⎥ ⎢ ⎥ ⎢ h h + 1 h + 2 h + 3 · · · h + v − 2 h + v − 1 h + v − 1 · · ·⎥ ⎢ ⎥ ⎢ h h + 1 h + 2 h + 3 · · · h + v − 2 h + v − 1 h + v − 1 · · ·⎥ ⎣ ⎦ .. .. .. .. . . . . 1 2 3 4 .. 2 3 4 5 .. 3 4 5 6 .. 4 5 6 7 .. ··· ··· ··· ··· v−1 v v+1 v+2 v v+1 v+2 v+3 .. v v+1 v+2 v+3 .. Proof. Let X = Xh,v . From the construction of X, we have αX = (v, 1, . . , 1) and h−1 βX = (h, 1, . . , 1). 30. In these cases, HX agrees with the statement.

14. Suppose λ = (4, 4, 3, 1, 1) 13. Then the Ferrers diagram of λ is •••• •••• ••• • • The conjugate of λ can be read off the Ferrers diagram of λ by counting the number of dots in each column as opposed to each row. In this example λ ∗ = (5, 3, 3, 2). 15. Let X ⊆ P1 × P1 be a finite set of reduced points. 9, the set of points looks like a Ferrers diagram. 16. 4. 9, the set of points X resembles the Ferrers diagram of the partition αX = (6, 5, 3, 1, 1). 5. One can deduce the following facts directly from the definitions.

Bh−1 s s .. ⎤ b1 · · · b2 · · ·⎥ ⎥ b3 · · ·⎥ ⎥ ⎥ ⎥ ⎥ ⎥. bh−1 · · ·⎥ ⎥ s · · ·⎥ ⎥ s · · ·⎥ ⎦ .. . . Proof. 27 (i) and (ii) because the left most column is the Hilbert function of h points in P1 and the top row is the Hilbert function of v points in P1 . 17, we have h = a1 and v = b1 . The entries in columns j = 2, . . 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.

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