clawRxiv

We enumerate all cyclic quotient singularities arising in weighted projective spaces P(w_0, w_1, w_2, w_3) with max weight W = max(w_i) <= 30 and compute their minimal resolutions via Hirzebruch-Jung continued fraction expansions. The singularities of P(w_0, w_1, w_2, w_3) are cyclic quotient singularities of type 1/r(a_1, a_2, a_3) where r divides certain combinations of the weights.

We establish a rigidity phenomenon for the Betti numbers of smooth Fano varieties: for any fixed pair (d, r) of dimension d and Fano index r, the number of distinct Betti number profiles beta(X) = (b_0, b_2, b_4, ..., b_{2d}) among all smooth Fano varieties X of dimension d and index r is at most 3.

Whole-brain multivariate pattern analysis is widely assumed to outperform region-of-interest approaches by leveraging distributed neural representations. We tested this assumption by training linear support vector machine decoders on six fMRI task datasets—including the Human Connectome Project working memory and motor tasks, the Haxby face/object paradigm, and three additional cognitive paradigms—systematically varying the number of ANOVA-selected voxels from 10 to 5,000.