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Given a sufficiently positive embedding $X\subset\mathbb{P}^N$ of a smooth projective variety $X$, we consider its secant variety $\Sigma$ that comes equipped with the embedding $\Sigma\subset\mathbb{P}^N$ by its construction. In this article, we determine the local cohomological dimension $\textrm{lcd}(\mathbb{P}^N,\Sigma)$ of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module $\mathcal{H}^{q}_{\Sigma}(\mathcal{O}_{\mathbb{P}^N})$, i.e., when $q=\textrm{lcd}(\mathbb{P}^N,\Sigma)$. Additionally, we show that $\Sigma$ has quotient singularities (in which case the equality $\textrm{lcd}(\mathbb{P}^N,\Sigma)=\textrm{codim}_{\mathbb{P}^N}(\Sigma)$ is known to hold) if and only if $X\cong\mathbb{P}^1$. We also provide a complete classification of $(X,L)$ for which $\Sigma$ has ($\mathbb{Q}$-)Gorentein singularities. As a consequence, we deduce that if $\Sigma$ is a local complete intersection, then either $X$ is isomorphic to $\mathbb{P}^1$, or an elliptic curve., Comment: 30 pages

We show that Veronese varieties of dimension $n \ge 4$ do not carry any Ulrich bundles of rank $r \le 3$. In order to prove this, we prove that a Veronese embedding of a complete intersection of dimension $m \ge 4$, which if $m=4$ is either $\mathbb P^4$ or has degree $d \ge 2$ and is very general and not of type $(2), (2,2)$, does not carry any Ulrich bundles of rank $r \le 3$., Comment: arXiv admin note: text overlap with arXiv:2405.01154

We characterize the existence of an Ulrich vector bundle on a variety $X \subset P^N$ in terms of the existence of a subvariety satisfying some precise conditions. Then we use this fact to prove that a complete intersection of dimension $n \ge 4$, which if $n=4$ is very general and not of type $(2,2)$, does not carry any Ulrich bundles of rank $r \le 3$ unless $n=4, r=2$ and $X$ is a quadric., Comment: v2: added link to a file with the Mathematica codes used to perform the calculations; v3: slightly improved statement and proof of Cor.1 (now sharp), updated some reference; v4: replaced a reference used in the proof of Thm. 1

We characterize $q$-ample Ulrich bundles on a variety $X \subseteq \mathbb P^N$ with respect to $(q+1)$-dimensional linear spaces contained in $X$.

We study the singularities of secant varieties of smooth projective varieties using methods from birational geometry when the embedding line bundle is sufficiently positive. More precisely, we study the Du Bois complex of secant varieties and its relationship with the sheaves of differential forms. Through this analysis, we give a necessary and sufficient condition for these varieties to have $p$-Du Bois singularities (in a sense that was proposed in [SVV23]). In addition, we show that the singularities of these varieties are never higher rational, by giving a classification of the cases when they are pre-1-rational. From these results, we deduce several consequences, including a Kodaira-Akizuki-Nakano type vanishing result for the reflexive differential forms of the secant varieties., Comment: 32 pages; v.2: Final version. Improved exposition and minor typos corrected, to appear in Proc. Lond. Math. Soc

We study varieties $X \subset P^r$ such that is $N_X^*(k)$ is an Ulrich vector bundle for some integer $k$. We first prove that such an $X$ must be a curve. Then we give several examples of curves with $N_X^*(k)$ an Ulrich vector bundle., Comment: v2: we specified the results that hold in characteristic zero and the ones holding in any characteristic. Thanks to crucial remarks of anonymous referee, we improved Theorem 3(ii) to $g \ge 3$ and we rewrote completely and shortened section 8 by using interpolation techniques

We study varieties $X \subseteq \mathbb P^N$ of dimension $n$ such that $T_X(k)$ is an Ulrich vector bundle for some $k \in \mathbb Z$. First we give a sharp bound for $k$ in the case of curves. Then we show that $k \le n+1$ if $2 \le n \le 12$. We classify the pairs $(X,\mathcal O_X(1))$ for $k=1$ and we show that, for $n \ge 4$, the case $k=2$ does not occur., Comment: 27 pages. v2: minor changes following referee's suggestions

We study the continuous CM-regularity of torsion-free coherent sheaves on polarized irregular smooth projective varieties $(X,\mathcal{O}_X(1))$, and its relation with the theory of generic vanishing. This continuous variant of the Castelnuovo-Mumford regularity was introduced by Mustopa, and he raised the question whether a continuously $1$-regular such sheaf $\mathcal{F}$ is GV. Here we answer the question in the affirmative for many pairs $(X,\mathcal{O}_X(1))$ which includes the case of any polarized abelian variety. Moreover, for these pairs, we show that if $\mathcal{F}$ is continuously $k$-regular for some integer $1\leq k\leq \dim X$, then $\mathcal{F}$ is a GV$_{-(k-1)}$ sheaf. Further, we extend the notion of continuous CM-regularity to a real valued function on the $\mathbb{Q}$-twisted bundles on polarized abelian varieties $(X,\mathcal{O}_X(1))$, and we show that this function can be extended to a continuous function on $N^1(X)_{\mathbb{R}}$. We also provide syzygetic consequences of our results for $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ on $\mathbb{P}(\mathcal{E})$ associated to a $0$-regular bundle $\mathcal{E}$ on polarized abelian varieties. In particular, we show that $\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$ satisfies $N_p$ property if the base-point freeness threshold of the class of $\mathcal{O}_X(1)$ in $N^1(X)$ is less than $\frac{1}{p+2}$. This result is obtained using a theorem in the Appendix written by Atsushi Ito., Comment: With an appendix by Atsushi Ito; v2: title changed, this is the second half of the previous submission, which includes the appendix. Final version, accepted for publication in Advances in Geometry

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., Comment: 19 pages, v2: minor changes following referee's suggestions. Final version, accepted for publication in Manuscripta Mathematica

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., Comment: 8 pages, Comments are welcome

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., Comment: 26 pages, changes made to the statement and proof of Lemma 2.10, minor changes made to the statement of Theorem 1.4, (1), (iii)

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., Comment: 36 pages, improved the results: they now apply to all surfaces of each family as opposed to a general surface of each family. Comments are welcome

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., Comment: 37 pages. New version considerably enlarged and reorganized; improved exposition; new, stronger results. Among other things, the title is changed, some sections and results have been split (e.g. Section 2 split into Sections 2, 4 and 5; (old) Section 4 split into Sections 6, 7 and 8; Thm. 3.5 split into Thms. 3.5, 3.7), new material and results have been added (e.g. Thm. 7.6; Sections 10 and 11)

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., Comment: Final version, accepted for publication in International Journal of Mathematics

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 $d=2$., Comment: 20 pages, final version, accepted for publication in Math. Z

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 hyperk\"ahler 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., Comment: 19 pages; improved previous results: title changed

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., Comment: 28 pages, final version, accepted for publication in J. Algebra