We find the defining equations of Rees rings of linearly presented height three
Gorenstein ideals. To prove our main theorem we use local cohomology techniques to
bound the maximum generator degree of the torsion submodule of symmetric powers
in order to conclude that the defining equations of the Rees algebra and of
the special fiber ring generate the same ideal in the symmetric algebra. We
show that the ideal defining the special fiber ring is the unmixed part of the
ideal generated by the maximal minors of a matrix of linear forms which
is annihilated by a vector of indeterminates, and otherwise has maximal
possible height. An important step in the proof is the calculation of the
degree of the variety parametrized by the forms generating the height three
Gorenstein ideal.
Keywords
blowup algebra, Castelnuovo–Mumford regularity, degree of a
variety, Hilbert series, ideal of linear type, Jacobian
dual, local cohomology, morphism, multiplicity, Rees ring,
residual intersection, special fiber ring