In this chapter we introduce bigraded rings and their properties, we introduce the biprojective space (Formula Presented) and its subvarieties, and we introduce the required background on Cohen-Macaulay rings. In particular, we build the correspondence between the bihomogeneous ideals of the bigraded ring R = k[x0, x1, y0, y1] and the varieties of (Formula Presented), mimicking the well-known correspondence between graded ideals of a polynomial ring and the varieties of (Formula Presented). While many of the results of this chapter extend quite naturally to any multiprojective space (Formula Presented), we will focus primarily on the case of (Formula Presented) (see the discussion at the end of the chapter for what is known in the general setting). Similarly, our discussion of Cohen-Macaulay rings takes place within the context of the bigraded ring R = k[x0, x1, y0, y1].
The biprojective space (Formula Presented)
Guardo E.;Van Tuyl A.
2015-01-01
Abstract
In this chapter we introduce bigraded rings and their properties, we introduce the biprojective space (Formula Presented) and its subvarieties, and we introduce the required background on Cohen-Macaulay rings. In particular, we build the correspondence between the bihomogeneous ideals of the bigraded ring R = k[x0, x1, y0, y1] and the varieties of (Formula Presented), mimicking the well-known correspondence between graded ideals of a polynomial ring and the varieties of (Formula Presented). While many of the results of this chapter extend quite naturally to any multiprojective space (Formula Presented), we will focus primarily on the case of (Formula Presented) (see the discussion at the end of the chapter for what is known in the general setting). Similarly, our discussion of Cohen-Macaulay rings takes place within the context of the bigraded ring R = k[x0, x1, y0, y1].I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
