In this paper we present an original, elementary way to compute the Hilbert–Poincare series of these rings; as a consequence we compute their Castelnuovo–Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo–Mumford regularity of a Cohen–Macaulay finitely generated graded module, we compute that of its Veronese transforms. | Turk J Math (2016) 40: 838 – 849 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Veronese transform and Castelnuovo–Mumford regularity of modules 1 Marcel MORALES1,2,∗, Nguyen Thi DUNG3 Institut Fourier, CNRS, Universit´e Joseph Fourier, Saint-Martin d’H`eres cedex, France 2 ESPE Universit´e de Lyon, France 3 Thai Nguyen University of Agriculture and Forestry, Thai Nguyen, Vietnam Received: • • Accepted/Published Online: Final Version: Abstract: Veronese rings, Segre embeddings, or more generally Segre–Veronese embeddings are very important rings in algebraic geometry. In this paper we present an original, elementary way to compute the Hilbert–Poincar´e series of these rings; as a consequence we compute their Castelnuovo–Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo–Mumford regularity of a Cohen–Macaulay finitely generated graded module, we compute that of its Veronese transforms. Key words: Castelnuovo–Mumford regularity, Veronese ring, Segre ring, Hilbert Series 1. Introduction Veronese rings, Segre embeddings, or more generally Segre–Veronese embeddings are very important rings in algebraic geometry. It is well known that these rings are arithmetically Cohen–Macaulay; hence their Hilbert– Poincar´e series can be written: PR (t) = QR (t) , (1−t)dim R where QR (t) is a polynomial on t with QR (1) ̸= 0 having positive integer coefficients; the sequence of the coefficients of QR (t) is also called the h− vector of R . The degree of QR (t) is the Castelnuovo–Mumford regularity (.[5][Chapter 4]), and the coefficient of the leading term of QR (t) is the highest graded Betti number of R . By using very original and elementary methods we are able to compute the leading term of QR (t). Our results allow to compute the Castelnuovo–Mumford regularity of the n Veronese module of any finitely generated .
