arXiv subjects

Anupam Bhatnagar

Publications and source records attributed to Anupam Bhatnagar.

4 recordsLinked to original sources

arXiv2014

The Moduli Space of Totally Marked Degree Two Rational Maps

A rational map $ϕ: \mathbb{P}^1 \to \mathbb{P}^1$ along with an ordered list of fixed and critical points is called a totally marked rational map. The space of totally marked degree two rational maps, $Rat^{tm}_2$ can be parametrized by an affine open subset of $(\mathbb{P}^1)^5$. We consider the natural action of $SL_2$ on $Rat^{tm}_2$ induced from the action of $SL_2$ on $(\mathbb{P}^1)^5$ and prove that the quotient space $Rat^{tm}_2/SL_2$ exists as a scheme. The quotient is isomorphic to a Del Pezzo surface with the isomorphism being defined over $\mathbb{Z}[1/2]$.

math.AG↗

arXiv2013

Isotriviality and the space of morphisms on projective varieties

Let $K=k(C)$ be the function field of a smooth projective curve $C$ over an infinite field $k$, let $X$ be a projective variety over $k$. We prove two results. First, we show with some conditions that a $K$-morphism $ϕ: X_K \to X_K$ of degree at least two is isotrivial if and only if $ϕ$ has potential good reduction at all places $v$ of $K$. Second, let $(X,ϕ), (Y,ψ)$ be dynamical systems where $X,Y$ are defined over $k$ and $g:X_{K} \to Y_{K}$ a dominant $K$-morphism, such that $g \circ ϕ= ψ\circ g$. We show under certain conditions that if $ϕ$ is defined over $k$, then $ψ$ is defined over $k$.

math.DS↗

arXiv2013

Projective varieties covered by isotrivial families

Let X,Y be projective schemes over a discrete valuation ring R, where Y is generically smooth and g:X \to Y a surjective R-morphism such that g_*\mathcal{O}_X = \mathcal{O}_Y. We show that if the family X \to Spec(R) is isotrivial, then the generic fiber of the family Y \to Spec(R) is isotrivial.

math.AG↗

arXiv2011

Very ample polarized self maps extend to projective space

Let $X$ be a projective variety defined over an infinite field, equipped with a line bundle $L$, giving an embedding of $X$ into $\mb{P}^m$ and let $ϕ: X \to X$ be a morphism such that $ϕ^*L \cong L^{\otimes q}, q\geq 2$. Then there exists an integer $r>0$ extending $ϕ^r$ to $\mb{P}^m$.

math.DS↗