Related papers: Lefschetz Theorem Using Real Morse Theory

We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature.…

We describe the hyperplane sections of the Severi variety of curves in $E \times \mathbb{P}^1$ in a similar fashion to Caporaso-Harris' seminal work. From this description we almost get a recursive formula for the Severi degrees (we get the…

We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small…

We show that the addition of points on the tropical Hesse curve can be realized via the intersection with a tropical line. Then the addition formula for the tropical Hesse curve is reduced from those for the level-three theta functions…

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by…

We show that if $X\subset\mathbb P^N_k$ is a normal variety of dimension $\geq 3$ and $H\subset\mathbb P^N_k$ a very general hypersurface of degree $d=4$ or $\geq 6$, then the restriction map $\mathrm{Cl}(X)\to\mathrm{Cl}(X\cap H)$ is an…

We prove a Lefschetz hyperplane theorem for the determinantal loci of a morphism between two holomorphic vector bundles $E$ and $F$ over a complex manifold under the condition that $E^*\ox F$ is Griffiths $k$-positive. We apply this result…

Let F be a polarized irreducible holomorphic symplectic fourfold, deformation equivalent to the Hilbert scheme parametrizing length-two zero-dimensional subschemes of a K3 surface. The homology group H^2(F,Z) is equipped with an integral…

For an arrangement of $n$ pseudolines in the real projective plane let us denote by $t_i$ the number of vertices incident to $i$ lines. We obtain a linear on $t_i$ inequality similar to the Hirzebruch one, but with an elementary proof. We…

We consider the homotopical dynamics on compact orientable surfaces of positive genus g. We establish a sufficient and necessary algebraic criterion for homotopy classes with infinitely many periodic points of maps on such surfaces in terms…

We show that the Grothendieck-Chow motive of a smooth hyperplane section $Y$ of an inner twisted form $X$ of a Milnor hypersurface splits as a direct sum of shifted copies of the motive of the Severi-Brauer variety of the associated cyclic…

Quantum Lefschetz theorem by Coates and Givental gives a relationship between the genus 0 Gromov-Witten theory of X and the twisted theory by a line bundle L on X. We prove the convergence of the twisted theory under the assumption that the…

We show the map $\sigma : T_g \to C_g$ sending a compact hyperbolic surface $X$ to a random simple closed geodesic on $X$ determines a proper embedding of Teichm\"uller space into the space of geodesic currents. The proof depends on a…

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this…

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main…

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and…

In this paper, we give the explicit bounds for the data of objects involved in some basic theorems of Singularity theory: the Inverse, Implicit and Rank Theorems for Lipschitz mappings, Splitting Lemma and Morse Lemma, the density and…

In this paper, we revisit the classical problem of determining osculating conics and sextactic points for a given algebraic curve. Our focus is on a particular family of plane cubic curves known as the Hesse pencil. By employing classical…

In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.

We introduce a notion of ampleness for subschemes of higher codimension using the theory of q-ample line bundles. We also investigate certain geometric properties satisfied by ample subvarieties, e.g. the Lefschetz hyperplane theorems and…