### The Waring loci of ternary quartics.

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Let Δ denote the discriminant of the generic binary -ic. We show that for d ≥ 3, the Jacobian ideal of Δ is perfect of height 2. Moreover we describe its SL-equivariant minimal resolution and the associated differential equations satisfied by Δ. A similar result is proved for the resultant of two forms of orders , whenever ≥ -1. If Φ denotes the locus of binary forms with total root multiplicity ≥ -n, then we show that the ideal of Φ is also perfect, and we construct a covariant which characterizes...

Let $A,B$ denote generic binary forms, and let ${\U0001d532}_{r}={(A,B)}_{r}$ denote their $r$-th transvectant in the sense of classical invariant theory. In this paper we classify all the quadratic syzygies between the $\left\{{\U0001d532}_{r}\right\}$. As a consequence, we show that each of the higher transvectants $\{{\U0001d532}_{r}:r\ge 2\}$ is redundant in the sense that it can be completely recovered from ${\U0001d532}_{0}$ and ${\U0001d532}_{1}$. This result can be geometrically interpreted in terms of the incomplete Segre imbedding. The calculations rely upon the Cauchy exact sequence of $S{L}_{2}$-representations, and the...

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