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Perhaps this should be attached to Charles Siegel's answer about the Hilbert scheme, but some concrete examples of degenerating flat families could be helpful. Some easy examples include conics turning into a fat line, skew lines colliding to produce an embedded point, and pairs of points on a line colliding to become fat. There are some nice relationships between these objects and families of constant coefficient linear differential equations via spectral schemes, e.g., the colliding points example says something about the behavior of solutions to $(\frac{d}{dz} - a)^2 - \lambda^2 = 0$ as $\lambda$ hits zero. I'm not sure what name gets attached to this relationship; I thought of "spectral schemes" but that name seems to be taken by a fancier notion.
Perhaps this should be attached to Charles Siegel's answer about the Hilbert scheme, but some concrete examples of degenerating flat families could be helpful. Some easy examples include conics turning into a fat line, skew lines colliding to produce an embedded point, and pairs of points on a line colliding to become fat. There are some nice relationships between these objects and families of constant coefficient linear differential equations, e.g., the colliding points example says something about the behavior of solutions to $(\frac{d}{dz} - a)^2 - \lambda^2 = 0$ as $\lambda$ hits zero. I'm not sure what name gets attached to this relationship; I thought of "spectral schemes" but that name seems to be taken by a fancier notion.