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We show that Hilbert schemes of points on supersingular Enriques surface incharacteristic 2 are simply connected, symplectic varieties but are notirreducible symplectic as the hodge number $h^{2,0} > 1$, even though asupersingular Enriques surface is an irreducible symplectic variety. These arethe classes of varieties which appear only in characteristic 2 and they showthat the hodge number formula for G\"ottsche-Soergel does not hold overcharacteristic 2. It also gives examples of varieties with trivial canonicalclass which are neither irreducible symplectic nor Calabi-Yau, thereby showingthat there are strictly more classes of simply connected varieties with trivialcanonical class in characteristic 2 than over $\mathbb{C}$ as given byBeauville-Bogolomov decomposition theorem.

Pathologies of Hilbert scheme of points of supersingular Enriques surface