Laudal’s toy model from e.g. p. 86 of this pdf comes to mind.

I've sent an e-mail to Bill Oxbury asking for a what license he would put on the code. I await a response.

I have a procedure called tanblowup written by Oxbury (shared with Stromme who shared it with me in 2002) for maple with schubert. Could that be of interest?

@DavidRydh Thank you! However, though all the files seem to be on archive.org between the blue captures, not all are on that link.

See H. Schubert (1879) p. 105 $\nu^9=92.$ Or this.

In M2 associatedPrimes yields $\langle {x}_{12}+{x}_{15},{x}_{11}-{x}_{16},{x}_{10}+{x}_{13},{x}_{9}-{x}_{14},{x}_{8}+{x}_{14},{x}_{7}-{x}_{13},{x}_{6}-{x}_{16},{x}_{5}+{x}_{15},{x}_{4}+{x}_{13},{x}_{3}+{x}_{14},{x}_{2}-{x}_{15},{x}_{1}-{x}_{16},{x}_{13}^{2}+{x}_{14}^{2}+{x}_{15}^{2}+{x}_{16}^{2}-1\rangle$.

Use this question with $d=14$ (see my answer to this question on MSE) to get a bound of 14784.

For real quartics, see Zeuthen cited from Plaumann, Sturmfels, Vinzant and summarised at Vinzant's page (archived).