In diophantine geometry, the Batyrev-Manin-Peyre conjecture originally concerns rational points on Fano varieties. It describes the asymptotic behaviour of the number of rational points of bounded height, when the bound becomes arbitrary large.
A geometric analogue of this conjecture deals with the asymptotic behaviour of the moduli space of rational curves on a complex Fano variety, when the 'degree' of the curves 'goes to infinity'. Various examples support the claim that, after renormalisation in a relevant ring of motivic integration, the class of this moduli space may converge to a constant which has an interpretation as a motivic Euler product.
In this talk, we will state this motivic version of the Batyrev-Manin-Peyre conjecture and give some examples for which it is known to hold : projective space, more generally toric varieties, and equivariant compactifications of vector spaces. In a second part we will introduce the notion of equidistribution of curves and show that it opens a path to new types of results.[-]

This talk will describe ongoing joint work with Paul Cadman and Duco van Straten, based on the PhD thesis of the former. Givental and Varchenko used the period mapping to pull back the intersection form on the Milnor fibre of an irreducible plane curve singularity $C$, and thereby define a symplectic structure on the base space of a miniversal deformation. We show how to combine this with a symmetric basis for the module of vector fields tangent to the discriminant, to produce involutive ideals $I_k$ which define the strata of parameter values $u$ such that $\delta(C_u)\leq k$. In the process we find an unexpected Lie algebra and a still mysterious canonical deformation of the module structure of the critical space over the discriminant. Much of this work is experimental - a crucial gap in understanding still needs bridging.[-]

In general the computation of the weight enumerator of a code is hard and even harder so for the coset leader weight enumerator. Generalized Reed Solomon codes are MDS, so their weight enumerators are known and its formulas depend only on the length and the dimension of the code. The coset leader weight enumerator of an MDS code depends on the geometry of the associated projective system of points. We consider the coset leader weight enumerator of $F_{q}$-ary Generalized Reed Solomon codes of length q + 1 of small dimensions, so its associated projective system is a normal rational curve. For instance in case of the $\left [ q+1,3,q-1 \right ]_{q}$ code where the associated projective system of points consists of the q + 1 points of a plane conic, the answer depends whether the characteristic is odd or even. If the associated projective system of points of a $\left [ q+1,4,q-2 \right ]_{q}$ code consists of the q + 1 points of a twisted cubic, the answer depends on the value of the characteristic modulo 6.[-]