Related papers: Irrelevant Exceptional Divisors for Curves on a Sm…
We initiate the study of computing shortest non-separating simple closed curves with some given topological properties on non-orientable surfaces. While, for orientable surfaces, any two non-separating simple closed curves are related by a…
We classify real families of minimal degree rational curves that cover an embedded rational surface. A corollary is that if the projective closure of a smooth surface is not biregular isomorphic to the projective closure of the unit-sphere,…
A formula for the dimension of the smoothing component of a $3$-dimensional isolated Cohen--Macaulay singularity is shown. We apply this formula for a $1$-convex threefold with a connected exceptional curve which is blown down to a terminal…
We classify invariant curves for birational surface maps that are expanding on cohomology. When the expansion is exponential, the arithmetic genus of an invariant curve is at most one. This implies severe constraints on both the type and…
We give obstructions - in terms of Gaussian maps - for a marked Prym curve $(C,\alpha,T_d)$ to admit a singular model lying on an Enriques surface with only one $d$-ordinary point singularity and in such a way that $T_d$ corresponds to the…
We give a method of counting the number of curves with a given type of singularity in a suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of singulaties for which our methods suffice include the…
We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X
We present a simple proof of the surface classification theorem using normal curves. This proof is analogous to Kneser's and Milnor's proof of the existence and uniqueness of the prime decomposition of 3-manifolds. In particular, we do not…
We study the systems of ordinary differential equations which are implicit with respect to the higher derivatives, appearing in the linear form, and their solutions near the singular points. The invertibility of the higher derivatives…
Consider a Hamiltonian diffeomorphism $g$ on a surface. We describe several scenarios where a curve $L$ and its image $g(L)$ provide a simple evidence that $g$ is not autonomous.
In this paper we study normal surfaces whose anticanonical divisors are strictly nef, i.e. (-K)C>0 for every curve C.
The article investigates the behaviour of the characteristic zero resolution invariant when transcribed suitably to the case of surfaces in positive characteristic. By Moh's jumping phenomenon -- or the occurrence of kangaroo singularities…
We prove that singular minimal surfaces with constant Gauss curvature are planes, spheres and cylindrical surfaces. We also classify all singular minimal surfaces with a constant principal curvature and singular minimal surfaces with…
Let $X$ be a semistable curve and $L$ a line bundle whose multidegree is uniform, i.e., in the range between those of the structure sheaf and the dualizing sheaf of $X$. We establish an upper bound for $h^0(X,L)$, which generalizes the…
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
Francesco Severi showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a…
We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in…
In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also,…
We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a…