We give an expression for the Smith-Thom deficiency of the Hilbert square $X^{ [2]}$ of a smooth real algebraic variety $X$ in terms of the rank of a suitable Mayer-Vietoris mapping in several situations. As a consequence, we establish a necessary and sufficient condition for the maximality of $X^{ [2]}$ in the case of projective complete intersections, and show that with a few exceptions no real nonsingular projective complete intersection of even dimension has maximal Hilbert square. We also provide new examples of smooth real algebraic varieties with maximal Hilbert square.
Categories: math.AG math.AT math.GT