We prove that quasi-projective base spaces of smooth families of minimal varieties of general type with maximal variation do not admit Zariski dense entire curves. We deduce the fact that moduli stacks of polarized varieties of this sort are Brody hyperbolic, answering a special case of a question of Viehweg and Zuo. For two-dimensional bases, we show analogous results in the more general case of families of varieties admitting a good minimal model.

We prove a conjecture of the first author relating the Bernstein-Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology. Contents 1. Introduction 1 2. The support of the specialization complex 3 3. Relative holonomic modules 6 4. Appendix 17 References 19

International audience Given positive integers $q$, $n$ and $d$, denote by $A_q(n,d)$ the maximum size of a $q$-ary code of length $n$ and minimum distance $d$. The famous Gilbert-Varshamov bound asserts that $A_q(n,d+1) \geq q^n / V_q(n,d)$, where $V_q(n,d)=\sum_{i=0}^d \binom{n}{i}(q-1)^i$ is the volume of a $q$-ary sphere of radius $d$. Extending a recent work of Jiang and Vardy on binary codes, we show that for any positive constant $\alpha$ less than $(q-1)/q$ there is a positive constant $c$ such that for $d \leq \alpha n, A_q(n,d+1) \geq c \frac{q^n}{ V_q(n,d)}n$. This confirms a conjecture by Jiang and Vardy.

For convex bodies K with C 2 boundary in R d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem.

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