Related papers: The Canonical Class of $\overline{M}_{0,n}(P^r,d)$…
This material is a rewriting and expansion of notes for beginning graduate students in seminars in combinatorics (Department of Mathematics, University of California San Diego). Solid skills in linear and multilinear algebra were required…
We develop various aspects of classical enumerative geometry, including Euler characteristics and formulas for counting degenerate fibres in a pencil, with the classical numerical formulas being replaced by identitites in the…
We classify the possible ramification data and etale local structure of orders over surfaces with canonical singularities.
We study the gonality and canonical model of a rational unicuspidal curve C. We are mainly interested in the case where C is non-Gorenstein. We classify such curves via different notions of gonality, and by its canonical model C', up to…
Jarnik's identity plays a major role in classical simultaneous approximation to two real numbers. O. German [2] has shown a generalization to the weighted setting in which the identity has to be replaced by two inequalities. His methods…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…
We use methods of the general theory of congruence and *congruence for complex matrices--regularization and cosquares-to determine a unitary congruence canonical form (respectively, a unitary *congruence canonical form) for complex matrices…
In this work we study, in greater detail than before, J.H. Conway's topographs for integral binary quadratic forms. These are trees in the plane with regions labeled by integers following a simple pattern. Each topograph can display the…
The equivalence group is determined for systems of linear ordinary differential equations in both the standard form and the normal form. It is then shown that the normal form of linear systems reducible by an invertible point transformation…
In the classification of real singularities by Arnold et al. (1985), normal forms, as representatives of equivalence classes under right equivalence, are not always uniquely determined. We describe the complete structure of the equivalence…
We develop general formulae for the numbers of conjugacy classes and irreducible complex characters of finite p-groups of nilpotency class less than p. This allows us to unify and generalize a number of existing enumerative results, and to…
We collect various known results (about plane curves and the moduli space of stable maps) to derive new recursive formulas enumerating low genus plane curves of any degree with various behaviors. Recursive formulas are given for the…
We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023,…
The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first…
In this article, we first describe codimension two regular foliations with numerically trivial canonical class on complex projective manifolds whose canonical class is not numerically effective. Building on a recent algebraicity criterion…
We compute all the Chern, Milnor and Pontryagin numbers for canonical toric manifolds associated with abstract simplicial complexes and the Stiefel-Whitney numbers for their real counterparts. Applications include combinatorial…
We found a regularity of the behavior of primes that allows to represent both prime and natural numbers as infinite matrices with a common formation rule of their rows. This regularity determines a new class of infinite cyclic groups that…
We present herewith certain thoughts on the important subject of nowadays physics, pertaining to the so-called ``singularities'', that emanated from looking at the theme in terms of ADG (: abstract differential geometry). Thus, according to…
We establish the canonical class inequality for families of higher dimensional projective manifolds. As an application, we get a new inequality between the Chern numbers of 3-folds with smooth families of minimal surfaces of general type…
We show that the resolution graph of a plane curve singularity admits a canonical decomposition into elementary graphs.