We study the canonical stability of a smooth projective 3-fold $V$ of general type. We prove that
(1) $|5K_V|$ gives a birational map onto its image provided the geometric genus $p_g\geq 4$;
(2) $|6K_V|$ gives a birational map provided $p_g=3$.
Known examples show that both are optimal. This fact can be viewed as parallel to surface case, though people know very little on 3-folds of general type with $p_g\leq 1$.
The definition of the weighted version of difficulty in "Termination of 4-fold canonical flips" contains some errors. In this paper, we describe these errors and how to correct them. Anyway, any sequence of 4-fold canonical flips terminates.
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.
We introduce K3 transitions as a geometric approach to studying canonical 3-folds. These transitions link different deformation classes of canonical 3-folds via a combination of birational contractions and smoothings. As applications, we investigate some basic properties of the web of canonical 3-folds in small codimension and give an interesting example of a singularity in codimension 4 with obstructed smoothings, similar to the famous example of the affine cone over a degree 6 del Pezzo surface.
We obtain that the nonzero accumulation points of the set of 3-fold canonical thresholds $\ct(X,S)$ are precisely $1/k$ where $k\ge 2$ is an integer and $S$ is an effective integral divisor of a projective 3-fold $X$ with only terminal singularities. Moreover, we generalize the ascending chain condition for the set of 3-fold canonical thresholds to pair.
T.Kishimoto raised the problem to classify all compactifications of contractible affine 3-folds into smooth Fano 3-folds with second Betti number two and classified such compactifications whose log canonical divisors are not nef. In this article, we show that there are 14 deformation equivalence classes of smooth Fano 3-folds which can admit structures of such compactifications whose log canonical divisors are trivial. We also construct an example of such compactifications with trivial log canonical divisors for each of all the 14 classes.
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…
Few explicit families of 3-folds are known for which the computation of the canonical ring is accessible and the birational geometry non-trivial. In this note we investigate a family of determinantal 3-folds in $\mathbb P^2 \times \mathbb P^3$ where this is the case.
We expand the theory of log canonical $3$-fold complements. We prove that if $X\rightarrow T$ is a $3$-dimensional contraction of log Calabi-Yau type, then we can find $B\geq 0$ on $X$ for which $(X,B)$ is log canonical and $n(K_X+B)\sim_T 0$, where $n$ is an uniform natural number. This means that every $3$-fold of log Calabi-Yau type can be turned into a log Calabi-Yau pair in an effective way.
We prove that the log canonical ring of a klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p>5$ is finitely generated. In the process we prove log abundance for such pairs in the case $κ=2$.
We prove that one can run the log minimal model program for log canonical $3$-fold pairs in characteristic $p>5$. In particular we prove the Cone Theorem, Contraction Theorem, the existence of flips and the existence of log minimal models for pairs with log divisor numerically equivalent to an effective divisor. These follow from our main results, which are that certain log minimal models are good.
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 the canonical volume of $V$. Assume $p_g\ge 2$. We show that $K^3\ge {1/3}$, which is a sharp lower bound. Then we classify those V with small $K^3$ up to explicit tructure. We also give some new examples with $p_g=2$ which have maximal canonical stability index. Finally we give an application to certain algebraic 4-folds.
Four-folds with trivial canonical bundles are divided into six classes according to their holonomy group. We consider examples that are fibred by abelian surfaces over the projective plane. We construct such fibrations in five of the six classes, and prove that there is no such fibration in the sixth class. We classify all such fibrations whose generic fibre is the Jacobian of a genus two curve.
For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $φ_{-m}$ is birational onto its image for all $m\geq 39$. By a weak $\mathbb{Q}$-Fano 3-fold $X$ we mean a projective one with at worst terminal singularities on which $-K_X$ is $\mathbb{Q}$-Cartier, nef and big. For weak $\mathbb{Q}$-Fano 3-folds, we prove that $φ_{-m}$ is birational onto its image for all $m\geq 97$.
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 $p_g=4$.
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 show that, for nonsingular projective 4-folds V of general type with geometric genus $p_g\geq 2$, the 33-canonical map is birational onto the image and the canonical volume has the lower bound $1/520$, which improves a previous theorem by Chen and Chen.
Let $V$ be a nonsingular projective 3-fold of general type. When the pluricanonical section index $δ(V)>12$, Chen-Chen \cite{Chen3} has a complete list of the possibility for the weighted basket ${\mathbb B}(V)$. However the possibility of ${\mathbb B}(V)$ could be infinite in the situation $δ(V)\leq 12$, which is the main challenge to the classification. In this paper we mainly study the case with $δ(V)=12$ and show that $Vol(V)\geq \frac{31}{48048}$, which improves the corresponding result of Chen--Chen \cite[Prop. 4.9(5)]{Chen3}.
We prove the abundance theorem for log canonical $n$-folds such that the boundary divisor is big assuming the abundance conjecture for log canonical $(n-1)$-folds. We also discuss the log minimal model program for log canonical $4$-folds.
By a canonical (resp. terminal) weak $\mathbb{Q}$-Fano $3$-fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is $\mathbb{Q}$-Cartier, nef and big. For a canonical weak $\mathbb{Q}$-Fano $3$-fold $V$, we show that there exists a terminal weak $\mathbb{Q}$-Fano $3$-fold $X$, being birational to $V$, such that the $m$-th anti-canonical map defined by $|-mK_{X}|$ is birational for all $m\geq 52$. As an intermediate result, we show that for any $K$-Mori fiber space $Y$ of a canonical weak $\mathbb{Q}$-Fano $3$-fold, the $…
The minimum of intersection numbers of the anti-canonical divisor with rational curves on a Fano manifold is called pseudo-index. It is expected that the intersection number of anti-canonical divisor attains to the minimum on an extremal ray, i.e. there exists an extremal rational curve whose intersection number with the anti-canonical divisor equals the pseudo-index. In this note, we prove this for smooth Fano 4-folds having birational contractions.
This paper aims to improve a theorem of Janos Kollar. For a given Complex projective threefold X of general type, suppose the plurigenus p_k(X)\ge 2, Kollar proved that the (11k+5)-canonical map is birational. Here we show that either the (7k+3)-canonical map or the (7k+5)-canonical map is birational and that the m-canonical map is stably birational for m\ge 13k+6. If P_k(X)\ge 3, then the m-canonical map is stably birational for m\ge 10k+8. In particular, the 12-canonical map is birational when p_g(X)\ge 2 and the 11-canonical map is birational when p_g(X)\ge 3.
In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, Δ)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $Δ$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+Δ\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, Δ)\in\mathfrak{D}\mbox{ for some }Δ\}$ is bounded from above.
This paper gives a simple example of a family of Calabi-Yaus of any dimension with canonical singularities of dimension one, whose Kuranishi space is singular. Thus the Bogomolov-Tian-Todorov unobstructedness theorem is not true for Calabi-Yaus with canonical singularities.
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 $P_{18}\geq 3$.
Motivated by previous computations of Y. Cao, M. Kool and the author, we propose square roots and sign rules for the vertex and edge terms that compute Donaldson-Thomas invariants of a toric Calabi-Yau 4-fold, and prove that they are canonical, exploiting the combinatorics of plane and solid partitions.
We construct some new deformation families of four-dimensional Fano manifolds of index $1$ in some known classes of Gorenstein formats. These families have explicit descriptions in terms of equations, defining their image under the anti-canonical embedding in some weighted projective space. The constructed families have relatively smaller anti-canonical degrees than most other known families of smooth Fano 4-folds.
A generic quartic 3-fold X admits a 7-dimensional family of representations as the Pfaffian of an 8 by 8 skew-symmetric matrix of linear forms. This provides a 7-dimensional moduli space M of rank 2 vector bundles on X. A precise geometric description of a 14-dimensional family of half-canonical curves C of genus 15 in X such that the above vector bundles are obtained by Serre's construction from C is given. It is proved that the Abel-Jacobi map of this family factors through M, and the resulting map from M to the intermediate Jacobian is quasi-finite. In particular, every component of M has n…
The notion of a d-critical locus is an ingredient in the definition of motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical locus structure on the Hilbert scheme of dimension zero subschemes on local toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we show the canonical d-critical locus structure has critical charts consistent with the description of Hilbert scheme as a degeneracy locus [BBS13]. In particular, the canonical d-critical locus structure is isomorphic to th…
We show that there exists a positive real number $δ>0$ such that for any normal quasi-projective $\mathbb{Q}$-Gorenstein $3$-fold $X$, if $X$ has worse than canonical singularities, that is, the minimal log discrepancy of $X$ is less than $1$, then the minimal log discrepancy of $X$ is not greater than $1-δ$. As applications, we show that the set of all non-canonical klt Calabi-Yau $3$-folds are bounded modulo flops, and the global indices of all klt Calabi-Yau $3$-folds are bounded from above.
In the first part of this article we show for some examples of surfaces of general type in toric 3-folds how to construct minimal and canonical models by toric methods explicitly. The examples we study turn out to be surfaces of general type, namely so called Kanev surfaces and Todorov surfaces. We show how properties of our examples of surfaces could be derived directly from properties of some polytopes and we compute the singularities of their canonical models.
A canonical Fano threefold has Fano index $\leq 66$
Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；School of Mathematics, Sun Yat-sen University（中山大学数学学院）
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 show that if $(X,B)$ is a log canonical pair with $\dim X\geq d+2$, whose non-klt centers have dimension $\geq d$, then $X$ is has depth $\ge d+2$ at every closed point.
Let $V$ be a complex nonsingular projective 3-fold of general type. We prove $P_{12}(V):=\text{dim} H^0(V, 12K_V)>0$ and $P_{m_0}(V)>1$ for some positive integer $m_0\leq 24$. A direct consequence is the birationality of the pluricanonical map $φ_m$ for all $m\geq 126$. Besides, the canonical volume $\text{Vol}(V)$ has a universal lower bound $ν(3)\geq \frac{1}{63\cdot 126^2}$.
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.
This note describes minimal surfaces $S$ of general type satisfying $p_g\geq 5$ and $K^2=2p_g$. For $p_g\geq 8$ the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if $p_g\geq 13$, $S$ has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for $p_g\leq 12$ there are two other classes of such surfaces with non birational canonical map.
In this paper, we classify projective toric birational morphisms from Gorenstein toric 3-folds onto the 3-dimensional affine space with relatively ample anti-canonical divisors.
We classify the irreducible components of the space of foliations on Fano 3-folds with rank one Picard group. As a corollary we obtain a classification of holomorphic Poisson structures on the same class of 3-folds.
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.
We show that for $5$-gonal curves of odd genus $g\geq 13$ and even genus $g\geq 28$ the $\lceil \frac{g-1}{2}\rceil$-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree $5$.
Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, ω_C)= \oplus_{k \geq 0} H^0(C, ω_C^k is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L)=\oplus_{k \geq 0} H^0(C,L^k)$ is generated in degree 1 if C is reduced with planar singularities and L is an invertible sheaf such that deg L_{|B} \geq 2p_a(B)+1 for every B \subseteq C.
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 $i\geq 0$.
