arXiv subjects

Jayan Mukherjee

Publications and source records attributed to Jayan Mukherjee.

14 recordsLinked to original sources

arXiv2026

Geography and Deformations of $\mathbb{Z}_2^s$-Covers of General Type Over Weighted Projective Threefolds

We study threefolds of general type constructed as $\mathbb{Z}_2^s$-covers of weighted projective spaces with a particular focus on their invariants, deformation theory, and the behavior of the $m$-canonical map. For the invariants, we write the ratios of the volume to the topological and holomorphic Euler characteristics as functions of the ratios of the degree of the branch divisors with respect to the total degree. From this expression, we obtain their asymptotic behavior, bounds, and a counterexample to a conjecture made by Bruce Hunt about the non-existence of smooth threefolds in a forbidden zone. From the perspective of deformation theory, we extend the criterion for such covers to be general in their moduli to the case when the weighted projective threefold has isolated singularities and the cover is non-flat, i.e., the pushforward of the structure sheaf splits as a direct sum of reflexive sheaves as opposed to line bundles. As an application, we present new numerical criteria for constructing components of the moduli spaces of stable threefolds and give concrete examples illustrating their application. Finally, we introduce techniques from Fourier transforms on finite groups to completely classify when a $\mathbb{Z}_2^s$-cover is a flat pluricanonical map. For $s \geq 2$, there are $32$ deformation types. We also show that there exist non-flat canonical and bicanonical $\mathbb{Z}_2^s$-covers for arbitrarily large values of $s$.

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arXiv2025

Extendability of general $K3$ surfaces without Gaussian maps and classification of non-prime Fano threefolds

In arXiv:2409.03960, we introduced an approach to the question of extendability of projective varieties via degeneration to ribbons. In this article we build on these methods to give a new proof of optimal results on the extendability of general non-prime $K3$ surfaces, classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime $K3$ surfaces for infinitely many values of $g$, for example when $g$ is of the form $g = 4k+1$, $k \geq 5$. This involves degenerations of $K3$ surfaces to ribbons on embedded Hirzebruch surfaces, called $K3$ carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the $K3$ carpets instead of computing coranks of Gaussian maps of the canonical curve sections. As a result of independent interest, we show such $K3$ carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension $6$ in the Hilbert scheme of $K3$ surfaces.

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arXiv2024

Syzygies of canonical ribbons on higher genus curves

We study the syzygies of the canonical embedding of a ribbon $\widetilde{C}$ on a curve $C$ of genus $g \geq 1$. We show that the linear series Clifford index and the resolution Clifford index are equal for a general ribbon of arithmetic genus $p_a$ on a general curve of genus $g$ with $p_{a} \geq \operatorname{max}\{3g+7, 6g-4\}$. Among non-general ribbons, the case of split ribbons is particularly interesting. Equality of the two Clifford indices for a split ribbon is related to the gonality conjecture for $C$ and it implies Green's conjecture for all double covers $C'$ of $C$ with $g(C') \geq \textrm{max}\{3g+2, 6g-4\}$. We reduce it to the vanishing of certain Koszul cohomology groups of an auxiliary module of syzygies associated to $C$, which may be of independent interest.

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arXiv2024

Extendability of projective varieties via degeneration to ribbons with applications to Calabi-Yau threefolds

In this article we study the extendability of a smooth projective variety by degenerating it to a ribbon. We apply the techniques to study extendability of Calabi-Yau threefolds $X_t$ that are general deformations of Calabi-Yau double covers of Fano threefolds of Picard rank $1$. The Calabi-Yau threefolds $X_t \hookrightarrow \mathbb{P}^{N_l}$, embedded by the complete linear series $|lA_t|$, where $A_t$ is the generator of Pic$(X_t)$, $l \geq j$ and $j$ is the index of $Y$, are general elements of a unique irreducible component $\mathscr{H}_l^Y$ of the Hilbert scheme which contains embedded Calabi-Yau ribbons on $Y$ as a special locus. For $l = j$, using the classification of Mukai varieties, we show that the general Calabi-Yau threefold parameterized by $\mathscr{H}_j^Y$ is as many times smoothly extendable as $Y$ itself. On the other hand, we find for each deformation type $Y$, an effective integer $l_Y$ such that for $l \geq l_Y$, the general Calabi-Yau threefold parameterized by $\mathscr{H}_l^Y$ is not extendable. These results provide a contrast and a parallel with the lower dimensional analogues; namely, $K3$ surfaces and canonical curves, which stems from the following result we prove: for $l \geq l_Y$, the general hyperplane sections of elements of $\mathscr{H}_l^Y$ fill out an entire irreducible component $\mathscr{S}_l^Y$ of the Hilbert scheme of canonical surfaces which are precisely $1-$ extendable with $\mathscr{H}^Y_l$ being the unique component dominating $\mathscr{S}_l^Y$. The contrast lies in the fact that for polarized $K3$ surfaces of large degree, the canonical curve sections do not fill out an entire component while the parallel is in the fact that the canonical curve sections are exactly one-extendable.

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arXiv2024

Projective smoothing of varieties with simple normal crossings

In this article, we introduce a new approach to show the existence and smoothing of simple normal crossing varieties in a given projective space. Our approach relates the above to the existence of nowhere reduced schemes called ribbons and their smoothings via deformation theory of morphisms. As a consequence, we prove results on the existence and smoothing of snc subvarieties $V \subset \mathbb{P}^N$, with two irreducible components, each of which are Fano varieties of dimension $n>2$, embedded inside $\mathbb{P}^{N}$ for effective values of $N$, by the complete linear series of a line bundle $H$. The general fibers of the resulting one parameter families are either smooth Fano, Calabi-Yau or varieties of general type, depending on the positivity of the canonical divisor of their intersections. An interesting consequence of projective smoothing is that it automatically gives a smoothing of the semi-log-canonical (slc) pair $(V, \Delta)$, where $\Delta = cH$, $c < 1$, is a rational multiple of a general hyperplane section of $H$. For threefolds, we are able to give explicit descriptions of the smoothable snc subvarieties due to the classification results of Iskovskikh-Mori-Mukai. In particular, we show the existence of unions V = $Y_1 \bigcup_D Y_2 \subset \mathbb{P}^N$, where $Y_i$'s are smooth anticanonically (resp. bi-anticanonically) embedded Fano threefolds, intersecting along $D$, where $D$ is either a del-Pezzo surface or a $K3$ surface (resp. a smooth surface with ample canonical bundle) and their smoothing in $\mathbb{P}^N$ to smooth Fano or Calabi-Yau threefolds (resp. to threefolds with ample canonical bundle) for various values of $N$ between $10$ and $163$. In cases when the general fiber is a smooth Fano or Calabi-Yau threefold, one can choose $c$ such that $(V, \Delta)$ is a Calabi-Yau pair while in all cases $c$ can be chosen so that $(V, \Delta)$ is a stable pair.

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arXiv2022

Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces

Let $\mathcal{E}$ be a vector bundle on a smooth projective variety $X\subseteq\mathbb{P}^N$ that is Ulrich with respect to the hyperplane section $H$. In this article, we study the Koszul property of $\mathcal{E}$, the slope-semistability of the $k$-th iterated syzygy bundle $\mathcal{S}_k(\mathcal{E})$ for all $k\geq 0$ and rationality of moduli spaces of slope-stable bundles on Del Pezzo surfaces. As a consequence of our study, we show that if $X$ is a Del Pezzo surface of degree $d\geq 4$, then any Ulrich bundle $\mathcal{E}$ satisfies the Koszul property and is slope-semistable. We also show that, for infinitely many Chern characters ${\bf v}=(r,c_1, c_2)$, the corresponding moduli spaces of slope-stable bundles $\mathfrak{M}_H({\bf v})$ when non-empty, are rational, and thereby produce new evidences for a conjecture of Costa and Mir\'o-Roig. As a consequence, we show that the iterated syzygy bundles of Ulrich bundles are dense in these moduli spaces.

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arXiv2021

A note on stability of syzygy bundles on Enriques and bielliptic surfaces

In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable with respect to $L$ for any ample and globally generated line bundle $L$ on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex Enriques surfaces improves a result of Torres-L\'opez and Zamora by removing a condition on Clifford index. Together with the results of Camere and Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample and globally generated line bundle $L$ on any smooth minimal complex projective surface $X$ of Kodaira dimension zero.

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arXiv2021

Tautological families of cyclic covers of projective spaces

In this article, we study the existence of tautological families on a Zariski open set of the coarse moduli space parametrizing certain Galois covers over projective spaces. More specifically, let ($1$) $\mathscr{H}_{n.r.d}$ (resp. $M_{n,r,d}$) be the stack (resp. coarse moduli) parametrizing smooth simple cyclic covers of degree $r$ over the projective space $\mathbb{P}^n$ branched along a divisor of degree $rd \geq 4$, and ($2$) $\mathscr{H}_{1,3,d_1,d_2}$ (resp. $M_{1,3,d_1,d_2}$) be the stack (resp. coarse moduli) of smooth cyclic triple covers over $\mathbb{P}^1$ with $2d_1-d_2 \geq 4$ and $2d_2-d_1 \geq 4$. In the former case, we show that such a family exists if and only if $\textrm{gcd}(rd, n+1) \mid d$ while in the latter case, we show that it always exists. We further show that even when such a family exists, often it cannot be extended to the open locus of objects without extra automorphisms. The existence of tautological families on a Zariski open set of its coarse moduli can be interpreted in terms of rationality of the stack if the coarse moduli space is rational. Combining our results with known results on the rationality of the coarse moduli of points on $\mathbb{P}^1$ and the coarse moduli of plane curves, we determine the rationality of $\mathscr{H}_{1,r,d}$ (resp. $\mathscr{H}_{2,r,d}$) for $rd \geq 4$ (resp. $rd\geq 49$). On the other hand $\mathscr{H}_{1,3,d_{1},d_{2}}$ is unirational, and we show that its coarse moduli $M_{1,3,d_1,d_2}$ is unirational and fibred over a rational base by homogeneous varieties which are rational if $\textrm{char}(\mathbb{k}) = 0$. Our study is motivated by the work of Gorchinskiy and Viviani on the moduli of hyperelliptic curves.

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arXiv2021

Deformations and moduli of irregular canonical covers with $K^2=4p_g-8$

In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying $K_X^2 = 4p_g(X)-8$, for any even integer $p_g\geq 4$. These surfaces also have unbounded irregularity $q$. We carry out our study by investigating the deformations of the canonical morphism $\varphi:X\to \mathbb{P}^N$, where $\varphi$ is Galois of degree 4. These canonical covers are classified in by the first two authors into four distinct families. We show that any deformation of $\varphi$ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of $\varphi$ is two-to-one onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that with the exception of one family, the deformations of $X$ are unobstructed, and consequently, $X$ belongs to a unique irreducible component of the Gieseker moduli space, which we prove is uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality $p_g > 2q-4$, with an irreducible component that has a proper "quadruple" sublocus where the degree of the canonical morphism jumps up. The existence of jumping subloci is a contrast with the moduli of surfaces with $K_X^2 = 2p_g - 4$, studied by Horikawa. There is a similarity and difference to the moduli of curves of genus $g\geq 3$, for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational.

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arXiv2020

Construction of varieties of low codimension with applications to moduli spaces of varieties of general type

In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions $m$, $m \geq 3$) that are analogues of the moduli space of curves of genus $g > 2$ with respect to the behavior of the canonical map and its deformations. In many cases, the general elements of these components are canonically embedded and their codimension is in the range of Hartshorne's conjecture.

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arXiv2020

K3 carpets on minimal rational surfaces and their smoothings

In this article, we study K3 double structures on minimal rational surfaces $Y$. The results show there are infinitely many non-split abstract K3 double structures on $Y = \mathbb{F}_e$ parametrized by $\mathbb P^1$, countably many of which are projective. For $Y = \mathbb{P}^2$ there exist a unique non-split abstract K3 double structure which is non-projective (see Dr\'ezet's article in arXiv:2004.04921). We show that all projective K3 carpets can be smoothed to a smooth K3 surface. One of the byproducts of the proof shows that unless $Y$ is embedded as a variety of minimal degree, there are infinitely many embedded K3 carpet structures on $Y$. Moreover, we show any embedded projective K3 carpet on $\mathbb F_e$ with $e<3$ arises as a flat limit of embeddings degenerating to $2:1$ morphism. The rest do not, but we still prove the smoothing result. We further show that the Hilbert points corresponding to the projective K3 carpets supported on $\mathbb{F}_e$, embedded by a complete linear series are smooth points if and only if $0\leq e\leq 2$. In contrast, Hilbert points corresponding to projective K3 carpets supported on $\mathbb{P}^2$ and embedded by a complete linear series are always smooth. The results in a recent paper of Bangere, Gallego, and Gonz\'alez show that there are no higher dimensional analogues of the results in this article.

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arXiv2020

Smoothing of multiple structures on embedded Enriques manifolds

We show that given an embedding of an Enriques manifold of index $d$ in a large enough projective space, there will exist embedded multiple structures with conormal bundle isomorphic to the trace zero module of the universal covering map, the universal cover being either a hyperk\"ahler or a Calabi-Yau manifold. We then show that these multiple structures (also known as $d$-ropes) can be smoothed to smooth hyperk\"ahler or Calabi-Yau manifolds respectively. Hence we obtain a flat family of hyperk\"ahler (or Calabi-Yau) manifolds embedded in the same projective space which degenerates to an embedded $d$-rope structure on the given Enriques manifold of index $d$. The above shows that these $d$-rope structures on the embedded Enriques manifold are points of the Hilbert scheme containing the fibres of the above family. We show that they are smooth points of the Hilbert scheme when $d=2$.

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arXiv2019

Remarks on projective normality for certain Calabi-Yau and hyperkähler varieties

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth fourfold with trivial canonical bundle, $A^{\otimes 15}$ is projectively normal for $A$ ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (upto deformation) of projective hyperkähler varieties. As a corollary we show that excepting two extremal cases in dimensions $4$ and $6$, a general curve section of any ample and globally generated linear system on the above mentioned examples is non-hyperelliptic.

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arXiv2019

On The Projective Normality And Normal Presentation On Higher Dimensional Varieties With Nef Canonical Bundle

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.

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