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R. V. Gurjar

Publications and source records attributed to R. V. Gurjar.

14 recordsLinked to original sources
arXiv2026

Derivations of plane algebroid curves

In this paper, we study the derivations of an irreducible plane algebroid curve $R:=\frac{k[[X,Y]]}{(f)}$, which is not regular. Since the normalization of $R$ is isomorphic to $k[[t]]$, every $k$-derivation of $R$ is induced from a $k$-derivation of $k[[t]]$, which are of the form $a(t)\frac{d}{dt}$ for some $a(t)\in k[[t]]$. We establish a lower bound on the $t$-order of a nonzero power series $a(t)$ for $a(t)\frac{d}{dt}$ to induce a derivation of $R$ in terms of the multiplicity of $R$. We also prove a related result for the value semi-group of such curves.

math.AC↗
arXiv2023

Invariants of Surfaces of Degree $d$ in $\mathbb{P}^n$

We prove that several invariants of a possibly singular complex affine or projective variety of degree $d$ in the affine space $\mathbb{A}^{n}$, or $\mathbb{P}^n$, are bounded by a function of $d$ alone, provided $b_{1}=0$ for a resolution of singularities of the variety.

math.AG↗
arXiv2017

A Note on Rational Cuspidal curves on $\mathbb{Q}$-Homology Projective Planes

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under suitable assumptions. We also provide examples which demonstrate sharpness of the results. The ambient surface is singular in these examples.

math.AG↗
arXiv2015

Fundamental Group of some Genus-2 Fibrations and Applications

We will prove that given a genus-2 fibration $f: X \rightarrow C$ on a smooth projective surface $X$ such that $b_1(X)=b_1(C)+2$, the fundamental group of $X$ is almost isomorphic to $π_1(C) \times π_1(E)$, where $E$ is an elliptic curve. We will also verify the Shafarevich Conjecture on holomorphic convexity of the universal cover of surfaces $X$ with genus-2 fibration $X\rightarrow C$ such that $b_1(X)>b_1(C)$.

math.AG↗
arXiv2015

Affine threefolds admitting $G_a$-actions

Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by $\mathbb{G}_a$-actions. We consider $\mathbb{G}_a$-actions on affine threefolds and discuss the structure and the singularities of the quotient surface as well as the singular fibers of the quotient morphism. Relative $\mathbb{G}_a$ and $\mathbb{G}_m$-actions on $\mathbb{A}^4$ or on an affine 4-fold having $\mathbb{A}^3$-fibrations over $\mathbb{A}^1$ are discussed when the actions leave one variable invariant or the fibration invariant.

math.AG↗
arXiv2014

Deformations of A^1-fibrations

We consider log deformations of affine surfaces with fibrations by the affine lines. Such a fibration is of affine type (resp. of complete type) if the base curve of the fibration is an affine curve (resp. a complete curve). The case of affine type is mainly treated in the article.

math.AG↗
arXiv2014

On the Zariski-Lipman conjecture for normal algebraic surfaces

We consider the Zariski-Lipman Conjecture on free module of derivations for algebraic surfaces. Using the theory of non-complete algebraic surfaces, and some basic results about ruled surfaces, we will prove the conjecture for several classes of affine and projective surfaces.

math.AG↗
arXiv2012

$\mathbb{A}^1_*$-fibrations on affine threefolds

The affine line minus one point is the underlying space of the algebraic torus of dimension one. However the fibration of an affine algebraic threefold by the affine line minus one point is not always the quotient morphism of the threefold by the algebraic torus. We investigate the structure of affine algebraic threefolds having fibrations by the affine line minus one point. Our results cover not only algebro-geometric but also topological properties of the threefolds. We also consider when the fibration by the affine line minus one point becomes the quotient morphism by an algebraic torus of dimension one.

math.AG↗
arXiv2012

Surjective derivations in small dimensions

Inspired by a result of D. Cerveau on surjective derivations on a polynomial ring in two variables over a complex number field C, we consider a surjective derivation defined on an affine domain over C of dimension one or two. Though our proofs are mostly algebraic or algebro-geometric, the idea using a result of Dimca-Saito which is behind the arguments of Cerveau and based on the differential complex of the polynomial ring in n variables is inspiring and affects our arguments.

math.AG↗
arXiv2001

On Gorenstein Surfaces Dominated by P^2

In this paper we prove that a normal Gorenstein surface dominated by the projective plane P^2 is isomorphic to a quotient P^2/G, where G is a finite group of automorphisms of P^2 (except possibly for one surface V_8'). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

math.AG↗
arXiv1995

Open problems on open algebraic varieties

This report records a large number of open problems in Affine Algebraic Geometry that were proposed by participants in a Conference on Open Algebraic Varieties at the Centre de Recherches en Mathematiques in Montreal at December 1994.

alg-geom↗