Hyperbolicity of general deformations

We present two methods of constructing low degree Kobayashi hyperbolic hypersurfaces in the projective space: the projection method and the deformation method. The talk is based on joint works of the speaker with B. Shiffman and C. Ciliberto.Comment: This is the content of the author's talk given at the conference "Effective Aspects of Complex Hyperbolic Varieties", Aber Wrac'h, France, September 10-14, 200

Algebraic invariants, mutation, and commensurability of link complements

We construct a family of hyperbolic link complements by gluing tangles along totally geodesic four-punctured spheres, then investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of nonisometric manifolds with the same volume and scissors congruence class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. All have trace field Q(i,\sqrt{2}), but some have integral traces while others do not.Comment: Minor changes following referee's suggestion

Closed Quasi-Fuchsian Surfaces In Hyperbolic Knot Complements

We show that every hyperbolic knot complement contains a closed quasi-Fuchsian surface.Comment: 69 pages, 27 figures. Made small changes suggested by refere

Plane curves with a big fundamental group of the complement

Let C \s \pr^2 be an irreducible plane curve whose dual C^* \s \pr^{2*} is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group \pi_1(\pr^2 \se C) contains a free group with two generators. If the geometric genus $g$ of $C$ is at least 2, then a subgroup of $G$ can be mapped epimorphically onto the fundamental group of the normalization of $C$, and the result follows. To handle the cases $g=0,1$, we construct universal families of immersed plane curves and their Picard bundles. This allows us to reduce the consideration to the case of Pl\"ucker curves. Such a curve $C$ can be regarded as a plane section of the corresponding discriminant hypersurface (cf. [Zar, DoLib]). Applying Zariski--Lefschetz type arguments we deduce the result from `the bigness' of the $d$-th braid group $B_{d,g}$ of the Riemann surface of $C$.Comment: 23 pages LaTeX. A revised version. The unnecessary restriction $d \ge 2g - 1$ of the previous version has been removed, and the main result has taken its final for

The hyperplanes of finite symplectic dual polar spaces which arise from projective embeddings

AbstractWe characterize the hyperplanes of the dual polar space DW(2n−1,q) which arise from projective embeddings as those hyperplanes H of DW(2n−1,q) which satisfy the following property: if Q is an ovoidal quad, then Q∩H is a classical ovoid of Q. A consequence of this is that all hyperplanes of the dual polar spaces DW(2n−1,4), DW(2n−1,16) and DW(2n−1,p) (p prime) arise from projective embeddings