Related papers: Jumping Numbers on Algebraic Surfaces with Rationa…
Based on high precision computation of periods and lattice reduction techniques, we compute the Picard group of smooth surfaces. We also study the lattice reduction technique that is employed in order to quantify the possibility of…
In this work, we refine a formula for the Tjurina number of a reducible algebroid plane curve defined over $\mathbb C$ obtained in the more general case of complete intersection curves in [1]. As a byproduct, we answer the affirmative to a…
Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of…
Given an ideal $a \subseteq R$ in a (log) $Q$-Gorenstein $F$-finite ring of characteristic $p > 0$, we study and provide a new perspective on the test ideal $\tau(R, a^t)$ for a real number $t > 0$. Generalizing a number of known results…
We study the family of rational curves on arbitrary smooth hypersurfaces of low degree using tools from analytic number theory.
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given. From this result, here we derive applications based on level curves to determine some…
In this paper, we give a formula for normal reduction number of an integrally closed $\mathfrak m$-primary ideal of a $2$-dimensional normal local ring $(A,\mathfrak m)$ in terms of the geometric genus $p_g(A)$ of $A$. Also we compute the…
A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven…
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…
In this paper the jump formulas for the double layer potential and other singular integrals are proved for arbitrary rectifiable sets, by defining suitable non-tangential limits. The arguments are quite straightforward and only require some…
Koll\'ar gave a series of examples of rational surfaces of Picard number $1$ with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up…
Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…
The aim of this work is to study duality of fractional ideals with respect to a fixed ideal and to investigate the relationship between value sets of pairs of dual ideals in admissible rings, a class of rings that contains the local rings…
In this paper we discuss the relationship between the moving planes of a rational parametric surface and the singular points on it. Firstly, the intersection multiplicity of several planar curves is introduced. Then we derive an equivalent…
Perfect ideals $I$ of grade $3$ in a local ring $(R,\mathfrak{m},\Bbbk)$ can be classified based on multiplicative structures on $\text{Tor}^R_{\bullet}(R/I,\Bbbk)$. The classification is incomplete in the sense that it remains open which…
The notion of $p_g$-ideals for normal surface singularities has been proved to be very useful. On the other hand, the core of ideals has been proved to be very important concept and also very mysterious one. However, the computation of the…
We introduce in this article a new method to estimate the minimum distance of codes from algebraic surfaces. This lower bound is generic, i.e. can be applied to any surface, and turns out to be ``liftable'' under finite morphisms, paving…
We consider modifications, for example blow ups, of Mori dream spaces and provide algorithms for investigating the effect on the Cox ring, e.g. testing finite generation or computing an explicit presentation in terms of generators and…
Given an ideal J on a smooth variety in characteristic zero, we estimate the F-jumping numbers of the reductions of J to positive characteristic in terms of the jumping numbers of J and the characteristic. We apply one of our estimates to…
In this paper, we will give a uniform upper bound of the number of rational points of bounded height in non-singular curves by applying the global determinant method.