There are many examples of 3-folds of general type with $χ(\mathcal {O})=1$ found by Iano-Fletcher and Reid about twenty years ago. Iano-Fletcher has ever proved $P_{12}(X)\ge 1$ and $P_{24}(X)\ge 2$ for all minimal 3-folds $X$ of general type with $χ(\mathcal {O}_X)=1$. In this paper, we improve on Fletcher's method. Our main result is that $ϕ_m$ is birational onto its image for all $m\ge 54$. To prove this we establish a general theory on birationality for general 3-folds which improve known results. Also we will show $P_m\geq 1$ for all $m\geq 14$ and $P_{18}\geq 3$.
We prove that for every positive integer $m\geq 18(2^{9}\cdot 3^{7})!$ and every smooth projective 3-fold of general type X defined over complex numbers, $\mid mK_{X}\mid$ gives a birational rational map from X into a projective space.
Let X be a projective complex 3-fold, quasihomogeneous with respect to an action of a linear algebraic group. We show that X is a compactification of SL_2/G, G a discrete subgroup, or that X can be equivariantly transformed into the 3-dim. projective space, the quadric Q_3, or into certain quasihomogeneous bundles with very simple structure.
In the present work we classify the relatively minimal 3-dimensional quasihomogeneous complex projective varieties under the assumption that the automorphism group is not solvable. By relatively minimal we understand varieties X having at most Q-factorial terminal singularities and allowing an extremal contraction X to Y, where dim Y <3.
In this paper, we will construct new examples of derived equivalent Calabi--Yau 3-folds with Picard number greater than one. We also study their mirror Calabi--Yau manifolds and find that they are given by Schoen's fiber products of suitable rational elliptic modular surfaces.
In two recent papers, we described some Siegel modular threefolds which admit a weak Calabi--Yau model. Not all of them admit a {\it projective} model. The purpose of this paper is to exhibit criterions for the projectivity, to treat several examples, and to compute their Hodge numbers.
In this paper we study pluricanonical maps of minimal projective 3-folds of general type with geometric genus $1$, $2$ and $3$. We go in the direction pioneered by Enriques and Bombieri, and other authors, pinning down, for low projective genus, a finite list of exceptions to the birationality of some pluricanonical map. In particular, apart from a finite list of weighted baskets, we prove the birationality of $φ_{16}$, $φ_{6}$ and $φ_{5}$ respectively.
Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $X$ be an irreducible smooth projective curve of genus $g$ over $k$. Fix an integer $n \geq 2$, and let $S^n(X)$ be the $n$-fold symmetric product of $X$. In this article we find the $S$-fundamental group scheme and Nori's fundamental group scheme of $S^n(X)$.
Let $p$ and $q$ be odd prime numbers. In this paper we study non-abelian pq-fold regular covers of the projective line, determine algebraic models for some special cases and provide a general isogeny decomposition of the corresponding Jacobian varieties. We also give a classification and description of the one-dimensional families of compact Riemann surfaces as before.
Let $X$ be a smooth projective variety of dimension $5$ and $L$ be an ample line bundle on $X$ such that $L^5>7^5$ and $L^d\cdot Z\geq 7^d$ for any subvariety $Z$ of dimension $1\leq d\leq 4$. We show that $\mathcal{O}_X(K_X+L)$ is globally generated.
We compute the expected dimension of the moduli space of torsion-free rank 2 sheaves at a point corresponding to a stable reflexive sheaf and give conditions for the existence of a perfect tangent-obstruction complex on a class of smooth projective threefolds; this class includes Fano and Calabi-Yau threefolds. We also explore both local and global relationships between moduli spaces of reflexive rank 2 sheaves and the Hilbert scheme of curves.
We construct new examples of Kobayashi hyperbolic hypersurfaces in the projective 4-space. They are generic projections of the triple symmetric product of a generic curve of genus at least 7, smoothly embedded in the projective 7-space.
We supply a detailed proof of the result by P.S. Green and T. H$\ddot{\text{u}}$bsch that all complete intersection Calabi--Yau 3-folds in product of projective spaces are connected through projective conifold transitions (known as the standard web). We also introduce a subclass of small transitions which we call primitive small transitions and study such subclass. More precisely, given a small projective resolution $π: \widehat{X} \rightarrow X$ of a Calabi--Yau 3-fold $X$, we show that if the natural closed immersion $Def(\widehat{X}) \hookrightarrow Def(X)$ is an isomorphism then $X$ has on…
As an application of universal polynomials for local and multi-singularities of maps, we revisit classical enumerative formulae of Salmon-Cayley-Zeuthen for projective surfaces and analogous formulae of Segre-(B.)Severi-Roth for projective $3$-folds. In particular, several examples of actual computation are given using universal polynomials for computing weighted Euler characteristics of singularity loci.
We address the following question: what are the cardinalities of maximal finite families of pairwise incident planes in a complex projective space? One proves easily that the span of the planes has dimension 5 or 6. Up to projectivities there is one such family spanning a 6-dimensional projective space - this is an elementary result. Maximal finite families of pairwise incident planes in a 5-dimensional projective space are considerably more misterious: they are linked to certain special (EPW) sextic hypersurfaces which have a non-trivial double cover, generically a hyperkaehler 4-fold. We pro…
Let $(X, A)$ be a nonsingular polarized toric 3-fold. We show that if the adjoint bundle of $A$ has no glabal section, then all ample line bundles on $X$ are normally generated. Even if the adjoint bundle is effective, if it is not big, then it is shown the normal generation. In particular, we show that all ample line bundles on a nonsingular toric Fano 3-fold are normally generated.
Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.
We will prove the following results for $3$-fold pairs $(X,B)$ over an algebraically closed field $k$ of characteristic $p>5$: log flips exist for $\Q$-factorial dlt pairs $(X,B)$; log minimal models exist for projective klt pairs $(X,B)$ with pseudo-effective $K_X+B$; the log canonical ring $R(K_X+B)$ is finitely generated for projective klt pairs $(X,B)$ when $K_X+B$ is a big $\Q$-divisor; semi-ampleness holds for a nef and big $\Q$-divisor $D$ if $D-(K_X+B)$ is nef and big and $(X,B)$ is projective klt; $\Q$-factorial dlt models exist for lc pairs $(X,B)$; terminal models exist for klt pair…
We construct Calabi-Yau 3-folds as orbifolds embedded in weighted projective space in codimension 4. For each Hilbert series that is realised, there are at least two different components of Calabi-Yau 3-folds.
In this paper, we prove that for a fibration $f:X\to Z$ from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field $k$ with $\mathrm{char} k =p >5$, if the geometric generic fiber $X_{\overlineη}$ is smooth, then subadditivity of Kodaira dimensions holds, i.e. $$κ(X)\geκ(X_{\overlineη})+κ(Z).$$
Nonsingular projective 3-folds $V$ of general type can be naturally classified into 18 families according to the {\it pluricanonical section index} $δ(V):=\text{min}\{m|P_m\geq 2\}$ since $1\leq δ(V)\leq 18$ due to our previous series (I, II). Based on our further classification to 3-folds with $δ(V)\geq 13$ and an intensive geometrical investigation to those with $δ(V)\leq 12$, we prove that $\text{Vol}(V) \geq \frac{1}{1680}$ and that the pluricanonical map $Φ_{m}$ is birational for all $m \geq 61$, which greatly improves known results. An optimal birationality of $Φ_m$ for the case $δ(V)=2$…
We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3\geq \frac{4}{3}p_g(X)-\frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.
Let $X$ be a projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.
Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that $H^j(X,Ω^i_X\otimes L)=0$ for every $j>0$, $i\geq 0$, and $L$ ample. This holds for toric varieties, but not for most other varieties. We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are many more than expected. Along the way, we conjecture that for every projective birational morphism $π\colon X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that is ample over $Y$, the higher direct image sheaf $R^jπ_*(Ω^i_X\otimes A)$ is zero for every $j>0$ and $i\geq 0$.
For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points.
Let $X$ be a nonsingular projective $n$-fold $(n\ge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $κ$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, \cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $κ$ is $0$, then the Chern ratios $(\frac{c_{2,1^{n-2}}}{c_{1^n}}, \frac{c_{2,2,1^{n-4}}}{c_{1^n}}, \cdots, \frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for al…
The conjectural Bogomolov-Gieseker (BG) type inequality for tilt semistable objects on projective 3-folds was proposed by Bayer, Macri and the author. In this note, we prove our conjecture for slope stable sheaves with the smallest first Chern class on certain Calabi-Yau 3-folds, e.g. quintic 3-folds.
Let $X$ be a Gorenstein minimal projective $n$-fold with at worst locally factorial terminal singularities, and suppose that the canonical map of $X$ is generically finite onto its image. When $n<4$, the canonical degree is universally bounded. While the possibility of obtaining a universal bound on the canonical degree of $X$ for $n \geqslant 4$ may be inaccessible, we give a uniform upper bound for the degrees of certain abelian covers. In particular, we show that if the canonical divisor $K_X$ defines an abelian cover over $\mathbb{P}^n$, i.e., when $X$ is an \emph{abelian canonical $n$-fol…
In this note we present two equivalent definitions for the genus of a birational map X --> Y between smooth complex projective 3-folds. The first one is the definition introduced in 1973 by M. A. Frumkin, the second one was recently suggested to me by S. Cantat. By focusing first on proving that these two definitions are equivalent, one can obtain all the results of the paper of Frumkin in a much shorter way. In particular, the genus of an automorphism of $\mathbb{C}^3$, view as a birational self-map of the projective space, will easily be proved to be 0.
We show that if an ample line bundle L on a nonsingular toric 3-fold satisfies h^0(L+2K)=0, then L is normally generated. As an application, we show that the anti-canonical divisor on a nonsingular toric Fano 4-fold is normally generated.
Let X be a smooth complex projective variety of dimension 4 and let L be an ample line bundle on X. In this paper, we study a natural number m such that h^{0}(m(K_{X}+L))>0 for any polarized 4-folds (X,L) with κ(K_{X}+L)\geq 0.
We prove the following results for projective klt pairs of dimension $3$ over an algebraically closed field of char $p>5$: the cone theorem, the base point free theorem, the contraction theorem, finiteness of minimal models, termination with scaling, existence of Mori fibre spaces, etc.
Let $\Cal E$ be a very ample vector bundle of rank two on a smooth complex projective threefold $X$. An inequality about the third Segre class of $\Cal E$ is provided when $K_X+\det \Cal E$ is nef but not big, and when a suitable positive multiple of $K_X+\det \Cal E$ defines a morphism $X\to B$ with connected fibers onto a smooth projective curve $B$, where $K_X$ is the canonical bundle of $X$. As an application, the case where the genus of $B$ is positive and $\Cal E$ has a global section whose zero locus is a smooth hyperelliptic curve of genus $\geq 2$ is investigated, and our previous res…
We prove that a birational morphism of projective 3-folds, over a field of characteristic zero, can be made toroidal by performing a sequence of blow ups of points and nonsingular curves above the domain and target.
We prove that there exists a positive number $C_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, the automorphism group $Aut(X)$ satisfies the inequality $\sharp{Aut}(X)\leq C_{n}\cdotμ(X,K_{X})$, where $μ(X,K_{X})$ is the volume of $X$ with respect to $K_{X}$.
We show that Fano 4-folds with Picard number 5 have Lefschetz defect 3 if and only if they are toric of combinatorial type K. We also find a characterization for such varieties in terms of Picard number of prime divisors. Moreover, we discuss classification results for 4-dimensional complex smooth projective varieties admitting some particular fiber type contractions.
Let $V$ be a complex nonsingular projective 3-fold of general type with $χ(ω_V)\geq 0$ (resp. $>0$). We prove that the m-canonical map $Φ_{|mK_V|}$ is birational onto its image for all $m\ge 14$ (resp. $\geq 8$). Known examples show that the lower bound $r_3=14$ (resp. $=8$) is optimal.
In this Part II, D(10.2), of D(10), we take D(10.1) (arXiv:1302.2054 [math.AG]) as the foundation to define the notion of $Z$-semistable morphisms from general Azumaya nodal curves, of genus $\ge 2$, with a fundamental module to a projective Calabi-Yau 3-fold and show that the moduli stack of such $Z$-semistable morphisms of a fixed type is compact. This gives us a counter moduli stack to D-strings as the moduli stack of stable maps in Gromov-Witten theory to the fundamental string. It serves and prepares for us the basis toward a new invariant of Calabi-Yau 3-fold that captures soft-D-string …
For smooth projective 3-folds of general type, we prove that the relative canonical stability $μ_s(3)\leq 8$. This is induced from our improved result of Kollár: the m-canonical map of a smooth projective 3-fold of general type is birational whenever $m\geq 5k+6$, provided $P_k(X)\geq 2$. The Q-divisor method is intensively developed to prove our results.
We prove that for a smooth projective irregular $3$-fold $X$ with $K_X\equiv 0$ and a nef and big divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 3$ and all $P\in \text{Pic}^0(X)$. We also use the same method to deal with $4$-folds, and prove that for a smooth projective irregular $4$-fold $X$ with $K_X\equiv 0$ and an ample divisor $L$ on $X$, $|mL+P|$ gives a birational map for all $m\geq 5$ and all $P\in \text{Pic}^0(X)$. These results are also optimal.
For nonsingular projective 3-folds $X$ (of general type) whose geometric genus $p_g=h^0(X, K_X)$ is $\geq 5$, the birationality of the fourth canonical map $φ_{4,X}=Φ_{|4K_X|}$ was characterized by D.-Q. Zhang and the first author in 2008. This paper aims at characterizing the birationality of $φ_{4,X}$ for those $X$ with $p_g=4$.
We relate Pandharipande-Thomas stable pair invariants on Calabi-Yau 3-folds containing the projective plane with those on the derived equivalent orbifolds via wall-crossing method. The difference is described by generalized Donaldson-Thomas invariants counting semistable sheaves on the local projective plane, whose generating series form theta type series for indefinite lattices. Our result also derives non-trivial constraints among stable pair invariants on such Calabi-Yau 3-folds caused by Seidel-Thomas twist.
We modify the conjectural Bogomolov-Gieseker type inequality introduced by Bayer, Macri and Toda to construct a family of geometric Bridgeland stability conditions on smooth projective 3-folds. We give an equivalent conjecture which needs to check these inequalities for a small class of tilt stable objects. We extend some of the techniques in Li's work for Fano 3-folds of Picard rank one to establish our modified Bogomolov-Gieseker type inequality conjecture for general Fano 3-folds.
Let $V$ be a smooth projective 3-fold of general type. Denote by $K^{3}$, a rational number, the self-intersection of the canonical sheaf of any minimal model of $V$. One defines $K^{3}$ as a canonical volume of $V$. The paper is devoted to proving the sharp lower bound $K^{3}\ge {1/420}$ which can be reached by an example: $X_{46}\subseteq \mathbb{P}(4,5,6,7,23)$.
We prove that there exists a positive integer $ν_{n}$ depending only on $n$ such that for every smooth projective $n$-fold of general type $X$ defined over {\bf C}, $\mid mK_{X}\mid$ gives a birational rational map from $X$ into a projective space for every $m\geq ν_{n}$. This theorem gives an affirmative answer to Severi's conjecture. The key ingredients of the proof are the theory of AZD which was originated by the aurhor and the subadjunction formula for AZD's of logcanoncial divisors.
If $X$ is a smooth complex projective 3-fold with ample canonical divisor $K$, then the inequality $K^3\ge {2/3}(2p_g-7)$ holds, where $p_g$ denotes the geometric genus. This inequality is nearly sharp. We also give similar, but more complicated, inequalities for general minimal 3-folds of general type. (A noether type of inequality lies, by all means, on the other side of the Miyaoka-Yau inequality from the geographical point of view.)
Let $X$ be a smooth projective threefold of Picard number one for which the generalized Bogomlov-Gieseker inequality holds. We characterize the limit Bridgeland semistable objects at large volume in the vertical region of the geometric stability conditions associated to $X$ in complete generality and provide examples of asymptotically semistable objects. In the case of the projective space and $ch^β(E)=(-R,0,D,0)$, we prove that there are only a finite number of nested walls in the $(α,s)$-plane. Moreover, when $R=0$ the only semistable objects in the outermost chamber are the 1-dimensional Gi…
We show that the usual sufficient criterion for a generic hypersurface in a smooth projective manifold to have the same Picard number as the ambient variety can be generalized to hypersurfaces in complete simplicial toric varieties. This sufficient condition is always satisfied by generic K3 surfaces embedded in Fano toric 3-folds.
