No Arabic abstract
We prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees were much larger, e.g., 248832 (for just the non-degenerate roots) via a famous general result of Khovanski. Our bound is sharp, allows real exponents, allows degeneracies, and extends to certain systems of n-variate fewnomials, giving improvements over earlier bounds by a factor exponential in the number of monomials. We also derive analogous sharpened bounds on the number of connected components of the real zero set of a single n-variate m-nomial.
We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.
We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr`i-Toda, such as $mathbb P^3$ or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in $X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.
We present an approach over arbitrary fields to bound the degree of intersection of families of varieties in terms of how these concentrate on algebraic sets of smaller codimension. This provides in particular a substantial extension of the method of degree-reduction that enables it to deal efficiently with higher-dimensional problems and also with high-degree varieties. We obtain sharp bounds that are new even in the case of lines in $mathbb{R}^n$ and show that besides doubly-ruled varieties, only a certain rare family of varieties can be relevant for the study of incidence questions.
A real morsification of a real plane curve singularity is a real deformation given by a family of real analytic functions having only real Morse critical points with all saddles on the zero level. We prove the existence of real morsifications for real plane curve singularities having arbitrary real local branches and pairs of complex conjugate branches satisfying some conditions. This was known before only in the case of all local branches being real (ACampo, Gusein-Zade). We also discuss a relation between real morsifications and the topology of singularities, extending to arbitrary real morsifications the Balke-Kaenders theorem, which states that the ACampo--Gusein-Zade diagram associated to a morsification uniquely determines the topological type of a singularity.
In this paper, we study the Nisnevich sheafification $mathcal{H}^1_{acute{e}t}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then $mathcal{H}^1_{acute{e}t}(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf $mathcal{H}^1_{acute{e}t}(G)$ with the sheaf of $mathbb{A}^1$-connected components of the classifying space ${rm B}_{acute{e}t}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of $mathbb{A}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.