arXiv subjects
Publications and source records attributed to Supravat Sarkar.
We show that if $X$ is a projective variety of dimension $\geq 2$ that is not a rational surface, then the blow up of $X$ at sufficiently many very general points has no nontrivial automorphism. Similar results were known before for $\mathbb{P}^2$ and $\mathbb{P}^3.$
If $X$ is a smooth projective variety of dimension $\geq 2$ and Picard rank $1$ on which every ample line bundle has at least $2$ linearly independent sections, we prove that syzygy bundle of any nontrivial globally generated line bundle is stable. We apply this to show stability of syzygy bundles in non-general type complete intersections in weighted projective spaces, and all complete intersections in projective spaces. This extends earlier results of Jiang-Ren, Coand{\u{a}} and others. We also show that in any dimension $\geq 2$, there is a smooth projective variety of Picard rank $1$ with an unstable syzygy bundle. This completely answers a question of Fulger-Langer.
Let $\mathfrak{g}=\mathfrak{g}(A)$ be the Borcherds-Kac-Moody Lie algebra (BKM LA) for a BKM Cartan matrix $A$ that is filled by negative integers. Fix a Cartan subalgebra $\mathfrak{h}$ of $\mathrm{g}$ and the classical cone of dominant integral weights $P^+\subset \mathfrak{h}^*$. The non-integrable simple highest weight $\mathfrak{g}$-modules $L(\mu)$'s widely studied were those by Naito ([Trans. Amer. Soc., 1995]), for $\mu$'s dot-linked to $P^+$-translates of sums $- \sum_{j\in J}\alpha_j$ of mutually orthogonal and imaginary simple roots $\alpha_j$'s. Recently, we computed weights of all highest weight $\mathfrak{g}$-modules $V$'s (over all BKM LA's), and character of $L(\rho)$ for Weyl vector $\rho$. These needed a family of ``integrable'' $L(\mu)$'s for $\mu$'s inside our novel signed-dominant-integral cone $P^{\pm}$ (which generalizes $P^+$). Pairings $\mu(\alpha_i^{\vee})\leq 0$ for $\mu\in P^{\pm}$ are multiples of $\frac{A_{ii}}{2}$ for all $i$. Nevertheless, $L(\mu)$ contains ``Chevalley-Serre relations'' $f_i^{\frac{2}{A_{ii}}{\mu(\alpha_i^{\vee})}+1}L(\mu)_{\mu}=0$, which seem to be previously unstudied and even in Naito's works. This paper initiates in rank-2, the study of module structures and maximal vectors (or Verma embeddings) in Verma covers $M(\mu)$'s of $L(\mu)$'s for $\mu\in P^{\pm}$. Our goal in this is to explore in weight spaces of those Vermas, the strictness, or else a uniform equality, of lower bounds by Kac and Kazhdan ([Adv. Math., 1979]) for count of linearly independent maximal vectors. We obtain presentations and characters of all $V$'s when Kac-Kazhdan equation has unique solution in the interior of root-cone. This builds on results of Kac and Kazhdan in crucial unique solution case.
We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{\'o}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.
We classify smooth projective varieties of Picard rank 2 which has two structures of blow-up of projective space along smooth subvarieties of different dimensions. This gives a characterization of the so called quadro-cubic Cremona transformation.
We describe the indecomposable components of the tangent bundle of the punctual Hilbert scheme of a smooth projective surface. As an application, we prove a recent conjecture about classification of products of punctual Hilbert schemes of smooth projective surfaces. We also determine when two products of symmetric powers of a smooth variety can be isomorphic. As a key step in our proof, we give a new characterization of abelian varieties, which states that in dimension $\geq 2$, a smooth complex projective variety whose tangent bundle is trivial upto a line bundle twist is an abelian variety.
If $X$ is a quasi-projective variety over a field $k$ and $ϕ$ a birational endomorphism of $X$ that is injective outside a closed subset of codimension $\geq 2$, we prove that $ϕ$ is an automorphism. This generalizes an old theorem of Ax and proves a conjecture of Miyanishi. A key step in our proof is a finiteness result on class groups, which is of interest in its own right.
In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary conditions for existence of determinantal representation. As an application, we show that for any integer $d \geq 1,$ there is an indecomposable vector bundle $E_d$ of rank $2$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ of $\mathbb{P}^2$ arise as the degeneracy loci of a pair of holomorphic sections of $E_d$, upto an automorphism of $\mathbb{P}^2$. We use this result to obtain a linear algebraic application.
We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of all the hyperplane sections of projectivized tangent bundle of projective space, hence describing hyperplane sections of a rational homogeneous manifold of Picard rank $2$. This also simplifies and extends recent results of Mazouni-Nagaraj in higher dimensions. We also compute the Chow ring of these hyperplane sections.
Given varieties $X, Y, W$ and dominant morphisms $ϕ:X\to Y$ and $f:X\to W$ such that $f$ is constant on fibres of $ϕ$ , we give sufficient conditions to guarantee that $f$ descends to a rational map or a morphism $Y\to W.$ We pay special attention to the case that the ground field has positive characteristic. This extends previous works of Aichinger and Das, who proved similar results for some classes of affine varieties.
We classify complex smooth projective surfaces whose punctual Hilbert scheme has a non-natural automorphism preserving the big diagonal. This completely answers a question raised by Belmans, Oberdieck and Rennemo, and extends previous works by Boissi{è}re-Sarti, Hayashi, Sasaki, Girardet and Wang. We reduce this to studying the existence of non-natural automorphisms of symmetric powers. We study this question for higher dimensional varieties too, giving some sufficient conditions guaranteeing every automorphism of a symmetric power to be natural. As a corollary, we characterize smooth projective surfaces of Kodaira dimension $\geq 1$ whose punctual Hilbert scheme has a non-natural automorphism, this time not assuming the automorphism preserves the big diagonal. We also address the question, when a smooth projective variety is determined up to isomorphism by its punctual Hilbert scheme.
We describe when two multiprojective bundles (fibre products of projective bundles over the same base) over projective spaces are isomorphic as abstract varieties. We also describe when two relative symmetric powers of projective bundles over projective spaces are isomorphic. Finally, we describe the automorphisms of multiprojective bundles and relative symmetric powers of projective bundles over projective spaces.
We study several properties of the symmetric power $S^mX$ of a smooth variety $X$. We describe the Picard and divisor class groups of $S^mX$ when $X$ is projective. We give a complete description of the stratification of $S^mX$ by iterated singular locus in terms of some combinatorial data regarding partitions of the integer $m.$ This gives a new viewpoint of a natural stratification of $S^mX$ by multiplicities.
Let $D$ be a reduced divisor in $\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism $f\colon Y\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor.
We show that some important classes of weak Fano $3$-folds of Picard rank $2$ do not satisfy Bott vanishing. Using this we show that any smooth projective $3$-fold $X$ of Picard rank $2$ with $-K_X$ nef which is the image of a projective toric variety is toric. This proves a special case of a conjecture by Ochetta-Wisniewski, extending a corresponding previous work for Fano $3$-folds. We also show that a weak Fano $3$-fold of Picard rank $2$ having an int-amplified endomorphism is toric. This proves a special case of a conjecture by Fakhrudding, Meng, Zhang and Zhong, extending corresponding previous work for Fano $3$-folds.
We study singularity of effective $\mathbb{Q}$-divisors on products of projective spaces of multidegree $(1,1...,1).$ This generalizes works of Bath, Musta{ţ}{ă} and Walther on singularity of square-free polynomials. We also give a lower bound on the log canonical threshold of a hypersurface in products of projective spaces.
Assuming Hartshorne's conjecture on complete intersections, we classify projective bundles over projective spaces which has a smooth blow up structure over another projective space. Under some assumptions, we also classify projective bundles over projective spaces which has a smooth blow up structure over some arbitrary smooth projective variety, not necessarily a projective space. We verify which of the globally generated vector bundles over projective space of first Chern class at most five has the property that their projectivisation has a smooth blow up structure, with no additional assumption. In the way, we get some new examples of varieties with both projective bundle and smooth blow up structures.
In this article, we explore the extremal contractions of several projective bundles over smooth Fano varieties of Picard rank $1$. We provide a whole class of examples of projective bundles with smooth blow-up structures, derived from the notion of drums which was introduced by Occhetta-Romano-Conde-Wiśniewski to study interaction with $\mathbb{C}^*$-actions and birational geometry. By manipulating projective bundles, we give a simple geometric construction of the rooftop flip, which was introduced recently by Barban-Franceschini. Additionally, we obtain analogues of some recent results of Vats in higher dimensions. The list of projective bundles we consider includes all globally generated bundles over projective space with first Chern class $2$. For each of them, we compute the nef and pseudoeffective cones.