Volume 22, Issue 3

The Malgrange–Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange–Galois groupoid of the Painlevé VI family including all parameters as new dependent variables. We conclude that it is the pseudogroup of transformations preserving the parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of a Painlevé VI equation depending analytically on the parameters does not satisfy any new partial differential equation (including derivatives with respect to parameters) which is not derived from Painlevé VI.

A Nikulin configuration is the data of 16 disjoint smooth rational curves on a K3 surface. According to results of Nikulin, the existence of a Nikulin configuration means that the K3 surface is a Kummer surface, moreover the abelian surface from the Kummer structure is determined by the 16 curves. In the paper [16], we constructed explicitly non-isomorphic Kummer structures on some Kummer surfaces. In this paper we generalize the construction to Kummer surfaces with a weaker restriction on the degree of the polarization and we describe some cases where the previous construction does not work.

It is shown that a smooth surface lying on a nodal quartic hypersurface in ℙ 4 is either regular or an elliptic conic bundle of degree 8. Furthermore, the latter configuration is shown to exist.

We consider a generic family of polynomial maps f := ( f 1 , f 2 ) : ℂ 2 → ℂ 2 with given supports of polynomials, and degree deg f := max(deg f 1 , deg f 2 ). We show that the (non-) properness of maps f in this family depends uniquely on the pair of supports, and that the set of isolated points in ℂ 2 ∖ f (ℂ 2 ) has a size of at most 6 deg f . This improves an existing upper bound (deg f – 1) 2 proven by Jelonek. Moreover, for each n ∈ ℕ, we construct a dominant map f as above, with deg f = 2 n + 2, and having 2 n isolated points in ℂ 2 ∖ f (ℂ 2 ). Our proofs are constructive and can be adapted to a method for computing isolated missing points of f . As a byproduct, we describe those points in terms of singularities of the bifurcation set of f .

We construct families of smooth functions H : ℝ n +1 → ℝ such that the Euclidean ( n + 1)-space is completely filled by not necessarily round hyperspheres of mean curvature H at every point.

Let X be a smooth complex projective variety. In 2002, Bridgeland [6] defined a notion of stability for the objects in 𝔇 b ( X ), the bounded derived category of coherent sheaves on X , which generalised the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on X and the geometry of the variety. We construct new stability conditions for surfaces containing a curve C whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of Stab( X ), the stability manifold of X .We then construct the moduli space M σ (ℴ X ) of σ -semistable objects of class [ℴ X ] in K 0 ( X ) after wall-crossing.

We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.

We extend previous results about Putinar’s Positivstellensatz for cylinders of type S × ℝ to sets of type S × ℝ r in some special cases, taking into account r and the degree of the polynomial with respect to the variables moving in ℝ r (this is to say, in the non-bounded directions). These special cases are in correspondence with the ones where the equality between the cone of non-negative polynomials and the cone of sums of squares holds. Degree bounds are provided.

Let J g be the Jacobian locus and let P g +1 be the Prym locus, in the moduli space A g of principally polarized abelian varieties of dimension g , for g ≥ 7. We study the extrinsic geometry of J g ⊂ P g +1 , under the inclusion provided by the theory of generalized Prym varieties introduced by Beauville. More precisely, we address the problem if certain geodesic curves, defined with respect to the Siegel metric of A g , starting at a Jacobian variety [ JC ] ∈ A g of a curve [ C ] ∈ M g and with direction ζ ∈ T [ JC ] J g , are locally contained in P g +1 . The result is that for a general JC , the answer is negative, if the rank k of ζ is such that k < Cliff C − 3, where Cliff C denotes the Clifford index of C .

We present a necessary and sufficient condition for a maximal curve, defined over the algebraic closure of a finite field, to be realised as an HKG-cover. We use an approach via pole numbers in a rational point of the curve. For this class of curves, we compute their Weierstrass semigroup as well as the jumps of their higher ramification filtrations at this point, the unique ramification point of the cover.