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Parnashree Ghosh

Publications and source records attributed to Parnashree Ghosh.

11 recordsLinked to original sources
arXiv2025

On $\mathbb{A}^1$-contractibility of certain simple birational extensions of affine spaces

Over a field of characteristic zero, up to isomorphism of varieties, affine spaces are the only smooth $\mathbb{A}^1$-contractible affine varieties in dimensions $\leqslant 2$. However, in dimensions $\geqslant 3$, examples of smooth $\mathbb{A}^1$-contractible affine varieties that are not isomorphic to affine spaces have been constructed in recent works of Dubouloz--Fasel and Dubouloz--Ghosh. In this paper, we consider a generalized class of smooth affine varieties containing these examples of $\mathbb{A}^1$-contractible varieties, given by $$ a(x_m)b(x_1,\ldots,x_{m-1})y+f(z,t)+x_m=0, $$ and investigate when such a variety is actually isomorphic to an affine space. We establish that, for large subfamilies of these varieties, being isomorphic to an affine space is equivalent to the rectifiability of the corresponding hyperplane embedding in $\mathbb{A}^{m+3}$; that is, there exists an automorphism of the ambient affine space carrying the hypersurface onto a coordinate hyperplane. Thus, our result naturally connects $\mathbb{A}^1$-contractibility of these varieties with the classical embedding problem for affine spaces in codimension one, and provides new evidence toward the Abhyankar--Sathaye conjecture. A key ingredient in our approach is the study of singular $\mathbb{A}^1$-contractible affine curves over fields of characteristic zero. We describe the possible singularities of such curves and obtain a generalization of the classical result of Lin--Zaidenberg for topologically contractible affine plane curves.

math.AC↗
arXiv2025

Algebraic families of higher dimensional $\mathbb{A}^{1}$-contractible affine varieties non-isomorphic to affine spaces

We construct algebraic families of smooth affine $\mathbb{A}^1$-contractible varieties of every dimension $n\geq 4$ over fields of characteristic zero which are non-isomorphic to affine spaces and potential counterexamples to the Zariski Cancellation Problem. We further prove that these families of varieties are also counter examples to the generalized Cancellation problem.

math.AG↗
arXiv2024

On embedding of linear hypersurfaces

Linear hypersurfaces over a field $k$ have been playing a central role in the study of some of the challenging problems on affine spaces. Breakthroughs on such problems have occurred by examining two difficult questions on linear polynomials of the form $H:=\alpha(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (i) Whether $H$ defines a closed embedding of $\mathbb{A}^{m+2}$ into $\mathbb{A}^{m+3}$, i.e., whether the affine variety $\mathbb{V}\subseteq \mathbb{A}^{m+3}_k$ defined by $H$ is isomorphic to $\mathbb{A}^{m+2}_k$. (ii) If $H$ defines a closed embedding $\mathbb{A}^{m+2}\hookrightarrow \mathbb{A}^{m+3}$ then whether $H$ is a coordinate in $D$. Question (i) connects to the Characterization Problem of identifying affine spaces among affine varieties; Question (ii) is a special case of the formidable Embedding Problem for affine spaces. In their earlier work the first two authors had addressed these questions when $\alpha$ is a monomial of the form $\alpha(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1, 1 \leqslant i \leqslant m$ and $F$ is of a certain type. In this paper, using $K$-theory and $\mathbb{G}_a$-actions, we address these questions for a wider family of linear varieties. In particular, we obtain certain families of higher dimensional hyperplanes $H$ satisfying the Abhyankar Sathaye conjecture on the Embedding problem. For instance, we show that when the characteristic of $k$ is zero, $F \in k[Z,T]$ and $H$ defines a hyperplane, then $H$ is a coordinate in $D$ along with $X_1, X_2, \dots, X_m$. Our results in arbitrary characteristic yield counterexamples to the Zariski Cancellation Problem in positive characteristic.

math.AG↗
arXiv2024

On the family of affine threefolds $a(x)y=F(x,z,t)$

In recent decades, linear affine threefolds have enabled researchers to solve some of the challenging problems on affine spaces. Koras-Russell threefolds, especially the Russell Cubic over $\mathbb{C}$ and Asanuma threefolds over a field of positive characteristic, are striking examples of such linear threefolds.In this paper, we apply tools from $K$-theory and theory of $\mathbb{G}_a$-actions to linear threefolds of the form $G:=a(X)Y-F(X,Z,T)\in k[X,Y,Z,T]$, over an arbitrary field $k$. We give some equivalent conditions for $G$ to be a hyperplane (i.e., $k[X,Y,Z,T]/(G)=k^{[3]}$) in the following cases: (i) $k$ is a field of characteristic zero (ii) $k$ is an arbitrary field and $a(X)$ has only multiple roots. We also establish the Abhyankar-Sathaye Conjecture affirmatively in these cases.

math.AG↗
arXiv2024

On characterization of Double Danielewski type algebras

Let $k$ be a field. In this paper, we consider Double Danielewski type algebras over an affine factorial $k$-domain $R$. We observe that this family produces a non-cancellative family of algebras over $R$. Further, when $k$ is a field of characteristic zero, we give a characterization for an affine algebra to be isomorphic to an algebra of Double Danielewski type.

math.AC↗
arXiv2023

On the triviality of an $\mathbb{A}^2$-fibration over a DVR

In this paper we show that any $\mathbb{A}^2$-fibration over a discrete valuation ring which is also an $\mathbb{A}^2$-form is necessarily a polynomial ring. Further we show that separable $\mathbb{A}^2$-forms over PIDs are trivial.

math.AC↗
arXiv2022

On Generalised Danielewski and Asanuma varieties

In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions ($\geqslant 2$) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the Makar-Limanov and Derksen invariant of generalised Asanuma varieties under certain hypotheses. We also establish a necessary and sufficient condition for certain generalised Asanuma varieties to be isomorphic to polynomial rings.

math.AC↗
arXiv2022

On the triviality of a family of linear hyperplanes

Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of $\mathbb{V}$ and $G= X_1^{r_1}\cdots X_m^{r_m}Y-F(X_1, \dots, X_m,Z,T)$. In \cite{com}, the second author had studied the case $m=1$ and had obtained several necessary and sufficient conditions for $\mathbb{V}$ to be isomorphic to the affine 3-space and $G$ to be a coordinate in $k[X_1, Y,Z,T]$. In this paper, we study the general higher-dimensional variety $\mathbb{V}$ for each $m \geqslant 1$ and obtain analogous conditions for $\mathbb{V}$ to be isomorphic to $\mathbb{A}^{m+2}$ and $G$ to be a coordinate in $k[X_1, \dots, X_m, Y,Z,T]$, under a certain hypothesis on $F$. Our main theorem immediately yields a family of higher-dimensional linear hyperplanes for which the Abhyankar-Sathaye Conjecture holds. We also describe the isomorphism classes and automorphisms of integral domains of the type $A$ under certain conditions. These results show that for each $d \geqslant 3$, there is a family of infinitely many pairwise non-isomorphic rings which are counterexamples to the Zariski Cancellation Problem for dimension $d$ in positive characteristic.

math.AC↗
arXiv2022

A note on homogeneous rank $2$ locally nilpotent derivations on $k[X,Y,Z]$

In this article we show that for every prime number $p$, any irreducible homogeneous locally nilpotent derivations of rank $2$ and degree $p-2$ are triangularizable. Further, we describe the structure of irreducible non-triangularizable homogeneous locally nilpotent derivations of rank $2$ and degree $pq-2$, where $p,q$ are prime numbers. Consequently, we give explicit descriptions of the generators of the image ideals of certain homogeneous locally nilpotent derivations of rank $2$.

math.AC↗
arXiv2022

Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$

Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In this paper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to the standard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be at most $2$ if $\deg(D) \leqslant 3$. As a consequence we obtain a certain class of homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ is a Dedekind domain, we give a bound for minimum number of generators of $\ker(D)$ as an $R$-algebra if $\deg(D) \leqslant 3$.

math.AC↗