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Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial. (English) Zbl 1333.13028
The authors look for the minimal possible Castelnuovo-Mumford regularity $$m_{p(z)}$$ of schemes with a given Hilbert polynomial $$p(z)$$ in characteristic $$0$$, providing a sharp lower bound. Many authors have focused their efforts in finding bounds for the Castelnuovo-Mumford regularity $$\mathrm{reg}(X)$$ of a scheme $$X$$. A very famous upper bound is due to G. Gotzmann [Math. Z. 158, 61–70 (1978; Zbl 0352.13009)] and coincides with the regularity of the saturated lexicographical ideal with a given Hilbert polynomial $$H(z)$$. The authors involve the regularity $$\varrho$$ of the Hilbert function $$H(z)$$ of a scheme $$X$$ and the general hyperplane section $$Z$$ of $$X$$, they obtain that if $$H(\rho -1) > p \,(\rho -1)$$ then $$\mathrm{reg}(X) >\rho +1$$, by using the relation $$\mathrm{reg}(X)=\max\{\mathrm{reg}(Z),\varrho+1\}$$ proved in [F. Cioffi et al., Collect. Math. 60, No. 1, 89–100 (2009; Zbl 1188.14020)]. Moreover the authors introduce a well-suited notion of minimal functions exploiting an idea of L. Roberts [in: Curves Semin. at Queen’s, Vol. 2, Kingston/ Can. 1981–82, Queen’s Pap. Pure Appl. Math. 61, Exp. F, 21 p. (1982; Zbl 0593.13009)]. So they show that there is a scheme $$X$$ having Hilbert polynomial $$p(z)$$ and the minimal Hilbert function with the smallest possible $$\varrho$$, which achieves the Castelnuovo-Mumford regularity $$m_{p(z)}$$ and obtain a formula for $$m_{p(z)}$$ depending on the Hilbert function of the hyperplane section. Their proofs are based on two new constructive methods,the first is called “ideal graft” of two schemes and the second “expanded lifting”. Both these methods exploit the notion of growth-height lexicographic Borel sets introduced by D. Mall [J. Pure Appl. Algebra 150, No. 2, 175–205 (2000; Zbl 0986.14002)], and developed also in [F. Cioffi et al., Discrete Math. 311, No. 20, 2238–2252 (2011; Zbl 1243.14007)]. In an Appendix the authors give an algorithm and its implementation to compute $$m_{p(z)}$$ and a Borel ideal defining a scheme with regularity $$m_{p(z)}$$.

##### MSC:
 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D02 Syzygies, resolutions, complexes and commutative rings 68W30 Symbolic computation and algebraic computation 11Y55 Calculation of integer sequences 14Q99 Computational aspects in algebraic geometry
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