10 coefficients determine a degree 3 homogenous polynomial in $k[x,y,z]$. I understand that there is a degree 12 polynomial in these coefficients, called the discriminant, with 2040 terms, which vanishes precisely when the curve is singular. I'm working on some generalizations to tropical geometry, and I would like to understand the classical case. Could someone point me to a reference where that shows how to construct this and check that it has the desired properties? I know that Gelfand, Karponov, and Zelevinski have an important book about discriminants, but glancing through the table of contents it looks like it has a lot of prerequisits and may not be as concrete as I would like. Google searching has been fruitless so far.

10 coefficients determine a degree 3 homogenous polynomial in $k[x,y,z]$. I understand that there is a degree 12 polynomial in these coefficients, called the discriminant, with 2040 terms, which vanishes precisely when the curve is singular. I'm working on some generalizations to tropical geometry, and I would like to understand the classical case. Could someone point me to a reference where that shows how to construct this and check that it has the desired properties? I know that Gelfand, Kapranov, and Zelevinski have an important book about discriminants, but glancing through the table of contents it looks like it has a lot of prerequisits and may not be as concrete as I would like. Google searching has been fruitless so far.