Decomposable Ternary Cubics

Abstract

Cubic forms in three variables are parametrised by points of a projective space P 9 . We study the subvarieties in this space defined by decomposable forms. Specifically, we calculate their equivariant minimal resolutions and describe their ideals invariant-theoretically.

Covariants of binary sextics and vector-valued Siegel modular forms of genus two

“…One calculates the action of the Hecke operator T 2 and finds In this section we illustrate the effectivity of our approach and show how one can use covariants to construct a basis for the 3-dimensional space S 4,16 and use this to calculate eigenvalues for the Hecke operators.…”

“…Note that its square Sym 2 χ 12,2 is the generator of the space S 24,4 . Similarly, in the case of weight (8,14) our form vanishes with order 2 along H 2 1 and by dividing by χ 10 we find a form of weight (8, 4) with character.…”

“…Dividing these forms by χ 10 leads to a 6-dimensional space of cusp forms of weight (4, 36). These forms restrict to multiples of the transpose of [e 4 3 ⊗ 3 , 0, 0, 0, 3 ⊗ e 4 3 ]. Thus there is a 5-dimensional subspace of forms of weight (4, 36) vanishing along H 2 1 and these all vanish with multiplicity ≥ 2 along H 2 1 .…”

“…The covariant corresponding to A [8,4] Moreover, the covariant associated to A [6,6] is an invariant equal to a 0 a 6 − 6 a 1 a 5 + 15 a 2 a 4 − 10 a 2 3 and coincides (up to a multiplicative scalar) with the invariant A ∈ I . The covariants of the action of SL(2, C) form a graded ring C. This ring can be identified with the ring of invariants…”

“…Thus there is a 5-dimensional subspace of forms of weight (4, 36) vanishing along H 2 1 and these all vanish with multiplicity ≥ 2 along H 2 1 . We divide again by χ 10 to get a 5-dimensional space of cusp forms of weight (4,26). All the forms in this space restrict to multiples of the transpose of [0, 0, e 4 2 ⊗ e 4 2 , 0, 0] along H 2 1 .…”