2 research outputs found
Toric fiber products versus Segre products
The toric fiber product is an operation that combines two ideals that are
homogeneous with respect to a grading by an affine monoid. The Segre product is
a related construction that combines two multigraded rings. The quotient ring
by a toric fiber product of two ideals is a subring of the Segre product, but
in general this inclusion is strict. We contrast the two constructions and show
that any Segre product can be presented as a toric fiber product without
changing the involved quotient rings. This allows to apply previous results
about toric fiber products to the study of Segre products.
We give criteria for the Segre product of two affine toric varieties to be
dense in their toric fiber product, and for the map from the Segre product to
the toric fiber product to be finite. We give an example that shows that the
quotient ring of a toric fiber product of normal ideals need not be normal. In
rings with Veronese type gradings, we find examples of toric fiber products
that are always Segre products, and we show that iterated toric fiber products
of Veronese ideals over Veronese rings are normal.Comment: 16 pages, v2: small improvements, to appear in Abhandlungen aus dem
Mathematischen Seminar der Universit\"at Hamburg, v3: final version, small
correction in proof of Lemma 1