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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.
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