Some effective estimates for André–Oort in Y(1)n

Let {X\subset Y(1)^{n}} be a subvariety defined over a number field {{\mathbb{F}}} and let {(P_{1},\ldots,P_{n})\in X} be a special point not contained in a positive-dimensional special subvariety of X. We show that if a coordinate {P_{i}} corresponds to an order not contained in a single exceptional Siegel–Tatuzawa imaginary quadratic field {K_{*}}, then the associated discriminant {|\Delta(P_{i})|} is bounded by an effective constant depending only on {\deg X} and {[{\mathbb{F}}:{\mathbb{Q}}]}. We derive analogous effective results for the positive-dimensional maximal special subvarieties.From the main theorem we deduce various effective results of André–Oort type. In particular, we define a genericity condition on the leading homogeneous part of a polynomial, and give a fully effective André–Oort statement for hypersurfaces defined by polynomials satisfying this condition.

The maximum likelihood degree of a very affine variety

We show that the maximum likelihood degree of a smooth very affine variety is equal to the signed topological Euler characteristic. This generalizes Orlik and Terao’s solution to Varchenko’s conjecture on complements of hyperplane arrangements to smooth very affine varieties. For very affine varieties satisfying a genericity condition at infinity, the result is further strengthened to relate the variety of critical points to the Chern–Schwartz–MacPherson class. The strengthened version recovers the geometric deletion–restriction formula of Denhamet al. for arrangement complements, and generalizes Kouchnirenko’s theorem on the Newton polytope for nondegenerate hypersurfaces.

Counting Packings of Generic Subsets in Finite Groups

A packing of subsets $\mathcal S_1,\dots,\mathcal S_n$ in a group $G$ is an element $(g_1,\dots,g_n)$ of $G^n$ such that $g_1\mathcal S_1,\dots,g_n\mathcal S_n$ are disjoint subsets of $G$. We give a formula for the number of packings if the group $G$ is finite and if the subsets $\mathcal S_1,\dots,\mathcal S_n$ satisfy a genericity condition. This formula can be seen as a generalization of the falling factorials which encode the number of packings in the case where all the sets $\mathcal S_i$ are singletons.