Related papers: A Non-Standard Bezout Theorem

In 1971, Zariski proposed some questions in Theory of Singularities. One of such problems is the so-called, nowadays, Zariski's multiplicity conjecture. In this work, we consider the version of this conjecture for families. We answer…

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of…

We study hyperbolicity for quasi-projective varieties where the boundary divisor consists of n+1 numerically parallel effective divisors on a complex projective variety of dimension n, allowing non-empty intersection. Under explicit local…

We prove the first inverse theorem for point--sphere incidence bounds over finite fields in dimensions $d \ge 3$, showing that near-extremality forces algebraic rigidity. While sharp upper bounds have been known for over a decade, the…

Let $C$ be an irreducible algebraic curve defined over a number field and inside an algebraic torus of dimension at least 3. We partially answer a question posed by Levin on points on $C$ for which a non-trivial power lies again on $C$. Our…

We prove the Zilber--Pink conjecture for curves in $Y(1)^n$ whose Zariski closure in $(\mathbb{P}^1)^n$ passes through the point $(\infty, \ldots, \infty)$, going beyond the asymmetry condition of Habegger and Pila. Our proof is based on a…

We prove a general likely intersections theorem, a counterpart to the Zilber-Pink conjectures, under the assumption that the Ax-Schanuel property and some mild additional conditions are known to hold for a given category of complex quotient…

We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th…

We develop a rigidity theory for frameworks in $\mathbb{R}^3$ which have two coincident points but are otherwise generic and only infinitesimal motions which are tangential to a family of cylinders induced by the realisation are considered.…

We give a relation between the existence of a Zariski decomposition and the behavior of the restricted volume of a big divisor on a smooth (complex) projective variety. Moreover, we give an analytic description of the restricted volume in…

Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi--Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an \'etale quotient of…

We prove a far-reaching strengthening of Szemer\'edi's regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such…

A series of Zariski pairs and four Zariski triplets were found by using lattice theory of K3 surfaces. There is a Zariski triplet of which one member is a deformation of another.

Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we…

The conventional integration theory on supermanifolds had been constructed so as to possess (an analog of) Stokes' formula. In it, the exterior differential d is vital and the integrand is a section of a fiber bundle of finite rank. Other,…

We provide several families of compact complex curves embedded in smooth complex surfaces such that no neighborhood of the curve can be embedded in an algebraic surface. Different constructions are proposed, by patching neighborhoods of…

Using currents with minimal singularities, we construct pointwise minimal multiplicities for a real pseudo-effective $(1,1)$-class $\alpha$ on a compact complex $n$-fold $X$, which are the local obstructions to the numerical effectivity of…

We introduce a new criterion which tests if a given decomposition of a given ternary form $T$ of even degree is unique. The criterion is based on the analysis of the Hilbert function of the projective set of points $Z$ associated to the…

In this paper we extend the arithmetic intersection theory of adelic divisors on quasiprojective varieties developed by X. Yuan and S. W. Zhang to cover certain adelic arithmetic divisors that are not nef nor integrable. The key concept…

We study a class of semialgebraic convex bodies called discotopes. These are instances of zonoids, objects of interest in real algebraic geometry and random geometry. We focus on the face structure and on the boundary hypersurface of…