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In these notes we introduce a numerical function which allows us to describe explicitly (and nonrecursively) the Betti numbers, and hence, the Hilbert function of a set $Z$ of three fat points whose support is an almost complete intersection (ACI) in $\mathbb{P}^1 \times \mathbb{P}^1.$ A nonrecursively formula for the Betti numbers and the Hilbert function of these configurations is hard to give even for the corresponding set of five points on a special support in $\mathbb P^2$ and we did not find any kind of this result in the literature. Moreover, we also give a criterion that allows us to characterize the Hilbert functions of these special set of fat points.
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참고문헌 (16)
참고문헌 신청
E. Carlini / 2017 / On the Hilbert function of general fat points in P1 × P1 / ArXiv preprint arXiv:1711.06193
M. V. Catalisano / 1991 / "Fat" points on a conic / Comm. Algebra 19(8):2153~2168
S. Cooper / Symbolic powers of codimension two Cohen-Macaulay ideals
G. Favacchio / 2018 / The Hilbert function of bigraded algebras in k[P1×P1] / Journal of Commutative Algebra
G. Favacchio / 2017 / The minimal free resolution of fat almost complete inter-sections in P1 × P1 / Canad. J. Math 69(6):1274~1291
M. V. Catalisano / 1991 / "Fat" points on a conic / Comm. Algebra 19(8):2153~2168
S. Cooper / Symbolic powers of codimension two Cohen-Macaulay ideals
G. Favacchio / 2018 / The Hilbert function of bigraded algebras in k[P1×P1] / Journal of Commutative Algebra
G. Favacchio / 2017 / The minimal free resolution of fat almost complete inter-sections in P1 × P1 / Canad. J. Math 69(6):1274~1291
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