A Generically Nonreduced Component for Hilbert Function (1,4,10,10)

We give a computer-assisted proof that the very-compressed locus with Hilbert function \((1,4,10,10)\) is the reduced support of a generically nonreduced irreducible component of \(\Hilb^{25}(\A^4)\) over an algebraically closed field of characteristic zero. The support is \(\A^4\times\Gr(10,20)\) and has dimension \(104\) . This supplies the case \(a=10\) omitted from Jelisiejew's published theorem for \((1,4,10,a)\) , \(a=6,7,8,9\) , and identifies the generic component throughout the interval in AIM Problem 31. The new computation gives 244 primary-obstruction quadrics in 46 normal variables over \(\F_3\) . Exact F4 rounds produce a positive pure leading power of every variable, proving that the normal obstruction algebra is zero-dimensional. A properness argument transfers this certificate to characteristic zero, where the published Białynicki–Birula criterion applies. The general obstruction framework and the four earlier cases are established prior work. We do not compute the generic nilpotent local algebra or claim absolute priority.