Geometric Invariant Theory

Progress in the theory of complex algebraic curves

“…In fact, by compactifying the quotient of ^ by Sp(2g, Z), Satake and Bailey- Borel [1966] give a compactification JK g of Jt g such that JK g is a projective variety and J# g -^g has codimension 2 in Jf g , showing in particular that there are no nonconstant holomorphic functions on J( g . MUMFORD. A third, and purely algebraic, construction of the moduli space of curves was given as an application by Mumford and Deligne of Mumford's "Geometric Invariant Theory" (Mumford [1965], DeligneMumford [1969]). Here the extra structure tacked on to a curve is exactly that of a particular projective embedding: namely, we consider curves C embedded in projective space P 5^-6 by the linear system of sections of the third power of the canonical bundle (w c ) 03 (the third power is chosen because it gives an embedding of any curve of genus > 2, and is the smallest power to do so).…”

Curves of genus 2 and Desargues configurations

“…One has only to remark that a degenerate Desargues configuration D π of the first kind (respectively 2 nd kind, respectively 3 rd kind) admits 4 (respectively 3, respectively 1) complete quadrangles. In the same way as we deduced Theorem 1.2 from Lemma 1.1 we obtain from this Recall that a binary form f (x, y) of degree 6 is called stable if f admits no root of multiplicity ≥ 3 and that the space M b 6 of stable binary sextics exists (see [7]). In this section, we present the construction of Stephanos (see [11] -in a slightly different set up) associating to every Desargues configuration D π a binary sextic J π in a canonical way.…”

A viable model of dynamical supersymmetry breaking in the hidden sector

“…Take a chiral gauge theory which is known to have D flat directions and no supersymmetric ground state, add a number of gauge singlet superfields equal to the number of flat directions, and add a tree level superpotential which couples the gauge singlets to the gauge invariant polynomials of chiral superfields which parametrize the flat directions [17], so that the only remaining classically flat directions have unbroken gauge symmetry. Most of the resulting models do not have a stable supersymmetry breaking ground state, but our example is not the unique DSB model which could serve as a supersymmetry breaking hidden sector.…”