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Ananya Pal

Publications and source records attributed to Ananya Pal.

5 recordsLinked to original sources
arXiv2026

Regular Borel subalgebras of the Lie Algebra of Aut($\mathbb{A}^2$)

In this paper, we find all regular Borel subalgebras of Lie(Aut($\mathbb{A}^2$)), i.e., maximal solvable subalgebras generated by homogeneous derivations with respect to the standard $\mathbb{Z}^2$-grading. It follows that a regular Borel subalgebra has derived length $2$ or $3$. We also describe isomorphism classes of such subalgebras. On the way, we give a combinatorial description of the regular Abelian subalgebras and the regular solvable subalgebras of Lie(Aut($\mathbb{A}^2$)).

math.RA↗
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 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 $H:=α(X_1,\dots,X_m)Y - F(X_1,\dots, X_m,Z,T)\in D:=k[X_1,\ldots,X_m, Y,Z,T]$: (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 $α$ is a monomial of the form $α(X_1,\ldots,X_m) = X_1^{r_1}\dots X_m^{r_m}$; $r_i>1, 1 \leqslant i \leqslant m$ 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 rigidity of Pham-Brieskorn surfaces

It is well known that, over an algebraically closed field $k$ of characteristic zero, for any three integers $a,b,c\geq 2$, any Pham-Brieskorn surface $B_{(a,b,c)}:= k[X,Y,Z]/(X^a + Y^b + Z^c)$ is rigid when at most one of $a,b,c$ is 2 and stably rigid when $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\leq 1$. In this paper we consider Pham-Brieskorn domains over an arbitrary field $k$ of characteristic $p\geq 0$ and give sufficient conditions on $(a,b,c)$ for which any Pham-Brieskorn domain $B_{(a,b,c)}$ is rigid. This gives an alternative approach to showing that there does not exist any non-trivial exponential map on $k[X,Y,Z,T]/(X^mY+T^{p^rq} + Z^{p^e})= k[x,y,z,t]$, for $m,q>1$, $p\nmid mq$ and $e>r\geq 1$, fixing $y$, a crucial result used in the paper "On the cancellation problem for the affine space $\mathbb{A}^3$ in characteristic $p$" by first author, to show that the Zariski Cancellation Problem (ZCP) does not hold for the affine $3$-space. We also provide a sufficient condition for $B_{(a,b,c)}$ to be stably rigid. Along the way we prove that for integers $a,b,c\geq 2$ with $gcd(a,b,c) = 1$ and for $F(Y)\in k[Y]$, the ring $k[X,Y,Z]/(X^aY^b + Z^c+ F(Y))$ is a rigid domain.

math.AG↗