Parallel Kustin-Miller Unprojection
 
 
Description:  Given a Gorenstein graded ring R and an ideal I of codimension 1, such that R/I is also Gorenstein, one can define a new graded ring, R_un, as the quotient of R[y] by an ideal J, obtained from I. This new graded ring is called the Kustin-Miller unprojection of R,I. Geometrically, this procedure produces a projective scheme Proj(R_un) birational to Proj(R) whose geometry can be studied from that of Proj(R). For this reason, Kustin-Miller unprojection has found many applications in algebraic geometry, for example in the birational geometry of Fano 3-folds, in the construction of K3 surfaces and Fano 3-folds in weighted projective space and in the study of Mori flips.

In this talk we describe recent joint work with S.A. Papadakis on the notion of parallel Kustin-Miller unprojection. The initial data for parallel Kustin-Miller unprojection consists of a Gorenstein graded ring, R, together with a finite set, {I_1,...,I_n}, of ideals of R of codimension 1, whose quotients, R/I_j, are Gorenstein and which satisfy mild extra assumptions. The end product of parallel Kustin-Miller unprojection, R_un, is defined as the quotient of R[y_1,...,y_n] by an ideal obtained from the initial data. Parallel Kustin-Miller unprojection can be seen has a series of n Kustin-Miller unprojections obtained from I_1,...,I_n, in any order. This extends the reach of the technique and enables the construction of interesting new algebraic varieties.
Area(s):
Date:  2009-01-13
Start Time:   14:30
Speaker:  Jorge Neves (CMUC/Mat. FCTUC)
Place:  5.5
Research Groups: -Algebra and Combinatorics
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