Search

This document contains notes from the lectures of Corti, Koll\'ar, Lazarsfeld, and Musta\c{t}\u{a} at the workshop ``Minimal and canonical models in algebraic geometry" at MSRI, Berkeley, April 2007. The lectures give an overview of the recent advances on canonical and minimal models of algebraic varieties obtained by Hacon--McKernan and Birkar--Cascini--Hacon--McKernan., Comment: 29 pages. Lecture notes from MSRI workshop

This preprint has been withdrawn. It is because I will never publish this preprint since everything has been contained in my new preprint: arXiv:0907.1506. Please refer to arXiv:0907.1506. Please do not cite this preprint any more., Comment: v2: many explanations were added, v3: this preprint has been withdrawn. everything has been contained in arXiv:0907.1506

This preprint has been withdrawn. It is because I will never publish this preprint since everything has been contained in my new preprint: arXiv:0907.1506. Please refer to arXiv:0907.1506. Please do not cite this preprint any more., Comment: v2: introduction was expanded, minor revisions, v3: this preprint has been withdrawn. everything has been contained in arXiv:0907.1506

Let X be the germ of a Gorenstein 3-fold singularity and C a smooth curve through it such that the general hyperplane section S of X containing D is DuVal of type E6 or E7. In this paper we obtain criteria for the existence of a terminal divisorial extremal neighborhood f: Y-->X contracting an irreducible divisor E onto C and classify all such neighborhoods., Comment: A section has been added classifying all divisorial extremal neighborhoods f:Y-->X such that if E is the f-exceptional divisor and C=f(E), the general hyperplane section S of X containg C is DuVal of type E6. The previous version classified only the E7 case. 20 pages

In this paper, we completely work out the log minimal model program for the moduli space of stable curves of genus three. We employ a rational multiple $\alpha\delta$ of the divisor $\delta$ of singular curves as the boundary divisor, construct the log canonical model for the pair $(\bar{\mathcal M}_3, \alpha\delta)$ using geometric invariant theory as we vary $\alpha$ from one to zero, and give a modular interpretation of each log canonical model and the birational maps between them. By using the modular description, we are able to identify all but one log canonical models with existing compactifications of $M_3$, some new and others classical, while the exception gives a new modular compactification of $M_3$., Comment: 35 pages, 6 figures

In this short note we show that the closed cone of moving curves of a smooth Fano-threefold is polyhedral. The proof relies on a famous result of Bucksom, Demailly, Paun and Peternell which says that the strongly movable cone is dual to the cone of pseudoeffective divisors. Finally, we relate the extremal rays of the cone of moving curves on a threefold with extremal rays in the cone of strongly movable curves on a birational model obtained by a Mori contraction., Comment: 12 pages, 1 figure. Version 2: corrected typo in theorem. Version 3: citation added

We give a functorial normal crossing compactification of the moduli of smooth marked cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification $M_{0,n} \subset \bar{M}_{0,n}$., Comment: 49 pages, 8 figures. v2: Added section 3 "Iitaka fibration for subvarieties of algebraic tori"

Awfully idiosyncratic lecture notes from CMI summer school in arithmetic geometry July 31-August 4, 2006. Does not include: rationality problems, techniques of the minimal model problem and much of the rest. Includes: Lecture 0: geometry and arithmetic of curves Lecture 1: Kodaira dimension and properties, rational connectendess, Lang's and Campana's conjectures. Lecture 2: Campana's program; Campana constellations framed in terms of b-divisors, to allow for a definition of Kodaira dimension directly on the base. A speculative notion of firmaments which may deserve further investigation, especially the arithmetic side. Lecture 3: the minimal model program: very short discussion of bend-and-break; even shorter discussion of finite generation and the existence of flip. Lecture 4: Vojta's conjectures, Campana's conjectures, and ABC., Comment: CMI summer school in arithmetic geometry, Gottingen, 2006. Numerous corrections following referee's report

We present the elementary properties of log canonical centers of log varieties., Comment: 8 pages, LaTeX

We prove that the canonical ring of a smooth projective variety is finitely generated.

We prove that the degree of Fano threefolds with terminal Q-factorial singularities and Picard number one is at most 125/2 and the bound is sharp., Comment: 23 pages, LATEX

Minimal log discrepancies (mld's) are related not only to termination of log flips, and thus to the existence of log flips but also to the ascending chain condition (acc) of some global invariants and invariants of singularities in the Log Minimal Model Program (LMMP). In this paper, we draw clear links between several central conjectures in the LMMP. More precisely, our main result states that the LMMP, the acc conjecture for mld's and the boundedness of canonical Mori-Fano varieties in dimension $\le d$ imply the following: the acc conjecture for $a$-lc thresholds, in particular, for canonical and log canonical (lc) thresholds in dimension $\le d$; the acc conjecture for lc thresholds in dimension $\le d+1$; termination of log flips in dimension $\le d+1$ for effective pairs; and existence of pl flips in dimension $\le d+2$. This also gives new proofs of some well-known and new results in the field in low dimensions: the acc conjecture holds for $a$-lc thresholds of surfaces; the acc conjecture holds for lc thresholds of 3-folds; termination of 3-fold log flips holds for effective pairs; and the existence of 4-fold pl flips holds., Comment: 41 pages

We formulate a comparison of minimal log discrepancies of a variety and its ambient space with appropriate boundaries in terms of motivic integration. It was obtained also by Ein and Musta\c{t}\v{a} independently., Comment: 10 pages; minor revision, particularly Remark 3.3 and Lemma 4.1

We prove that any sequence of 4-dimensional log flips that begins with a klt pair (X,D) such that -(K+D) is numerically equivalent to an effective divisor, terminates. This implies termination of flips that begin with a log Fano pair and termination of flips in a relative birational setting. We also prove termination of directed flips with big K+D. As a consequence, we prove existence of minimal models of 4-dimensional dlt pairs of general type, existence of 5-dimensional log flips, and rationality of Kodaira energy in dimension 4., Comment: 13 pages; a minor change in the proof of Thm.4.3

Let $U\subset \mathbb P^N$ be a projective variety which is not a cone and whose hyperplane sections are smooth Enriques surfaces. We prove that the degree of a $U$ is at most 32 and the bound is sharp., Comment: 18 pages, LaTeX

Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program., Comment: 14 pages

We study the cone of effective divisors and the total coordinate ring of wonderful varieties, with applications to their automorphism group. We show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings., Comment: Final version, to appear in Journal of Algebra

We study smoothings of Fano threefolds. We prove that the Picard number remains constant in the case of terminal Gorenstein singularities., Comment: 6 pages, LaTeX2e

Let X be a projective irreducible symplectic manifold and L a non trivial nef divisor on X. Assume that the nef dimension of L is strictly less than the dimension of X. We prove that L is semiample, Comment: 5 pages

We prove inversion of adjunction on log canonicity., Comment: 4 pages; minor typos in the proof of Lemma 3 corrected

We introduce a diophantine property of a log canonical algebra, and use it to describe the restriction of a log canonical algebra of general type to a log canonical center of codimension one., Comment: 22 pages, LaTeX

Let X be a surface with quotient singularities which admits a smoothing to the plane. We prove that X is a deformation of a weighted projective plane P(a^2,b^2,c^2), where a,b,c is a solution of the Markov equation a^2+b^2+c^2=3abc. We also prove a generalisation for del Pezzo surfaces of degree K^2 at least 5., Comment: 14 pages

We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their counterexamples on ruled surfaces in positive characteristic., Comment: 17 pages

A complete classification is presented of elliptic and K3 fibrations birational to certain mildly singular complex Fano 3-folds. Detailed proofs are given for one example case, namely that of a general hypersurface X of degree 30 in weighted P^4 with weights 1,4,5,6,15; but our methods apply more generally. For constructing birational maps from X to elliptic and K3 fibrations we use Kawamata blowups and Mori theory to compute anticanonical rings; to exclude other possible fibrations we make a close examination of the strictly canonical singularities of (X,(1/n)H), where H is the linear system associated to the putative birational map and n is its anticanonical degree., Comment: 24 pages

We prove the existence of flips in dimension n, contingent on the termination of real flips in dimension n-1.

We show the finite generation of certain invariant graded algebras defined on toric weak log Fano fibrations. These are the toric version of FGA algebras, recently introduced by Shokurov in connections to the existence of flips., Comment: 23 pages

The parametric degree of a rational surface is the degree of the polynomials in the smallest possible proper parametrization. An example shows that the parametric degree is not a geometric but an arithmetic concept, in the sense that it depends on the choice of the ground field. In this paper, we introduce two geometrical invariants of a rational surface, namely level and keel. These two numbers govern the parametric degree in the sense that there exist linear upper and lower bounds., Comment: 16 pages, 1 figure

We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.

We prove that there is a fixed constant r=r_n, such that if X is a variety of general type, then the rth pluricanonical map is birational.

Let X be a Fano manifold of pseudoindex i_X whose Picard number is at least two and let R be an extremal ray of X with exceptional locus Exc(R). We prove an inequality which bounds the length of R in terms of i_X and of the dimension of Exc(R) and we investigate the border cases. In particular we classify Fano manifolds X of pseudoindex i_X obtained blowing up a smooth variety Y along a smooth subvariety T such that dim T < i_X.

This paper continues the study of non-general type subvarieties begun in a joint paper with M.Schneider and A.Sommese (Int.L.Math. 10, 1999). We prove uniruledness of a projective manifold containing a submanifold not of general type whose normal bundle has positivity properties and moreover study the rational quotient. We also relate the fundamental groups and prove a cohomological criterion for a manifold to be rationally connected., Comment: 11 pages, final version, to appear in International Journal of Math

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds., Comment: 15 pages. v2: Completely rewritten paper. Structure theorem for the cone of nef curves proved in arbitrary dimension using results of Birkar, Cascini, Hacon and McKernan. To appear in Mathematische Zeitschrift

We revisit the nef curve cone theorem and use it to reprove the pseudo-effectivity of the second Chern classes for terminal weak Q-Fano varieties., Comment: 11 pages

The Kleiman-Mori cone plays important roles in the birational geometry. In this paper, we construct complete varieties whose Kleiman-Mori cones have interesting properties. First, we construct a simple and explicit example of complete non-projective singular varieties for which Kleiman's ampleness criterion does not hold. More precisely, we construct a complete non-projective toric variety $X$ and a line bundle $L$ on $X$ such that $L$ is positive on $\bar {NE}(X)\setminus \{0\}$. Next, we construct complete singular varieties $X$ with $NE(X)=N_1(X)\simeq \mathbb R^k$ for any $k$. These explicit examples seem to be missing in the literature., Comment: 8 pages

We totally classify the projective toric varieties whose canonical divisors are divisible by their dimensions. In Appendix, we show that Reid's toric Mori theory implies Mabuchi's characterization of the projective space for toric varieties., Comment: 6 pages

We construct minimal smooth models of coarse moduli schemes of rational Drinfeld modules of rank 4 as well as their smooth compactifications., Comment: 15 pages, 9 figures

We give a reduction of the conjecture that for terminal projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective. On the other hand, some effective non-vanishing results are obtained as applications of the pseudo-effectivity of the second Chern classes., Comment: 15 pages

Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the Kodaira dimension if the determinant bundle has sufficiently large Kodaira dimension. This is based on the study of the determinant bundle of a quotient of the cotangent bundle of a non-uniruled manifold: this bundle is always pseudo-effective. We apply this to study the universal cover of a projective manifold. Finally we prove the following: if the canonical bundle is numerically equivalent to an effective Q-divisor, then the Kodaira dimension is non-negative., Comment: Latex, 29 pages, appendix by Matei Toma added; rest unchanged

In this paper, we give explicit combinatorial descriptions for toric extremal contractions under the relative setting, where varieties do not need to be complete. Fujino's completion theorem is the key to the main result. As applications, we can generalize some of Mustata's results related to Fujita's conjecture on toric varieties for the relative case., Comment: 8 pages

We give a counterexample and some conclusions for effective non-vanishing of Weil divisors on algebraic surfaces., Comment: 7 pages

The paper explores the birational geometry of terminal quartic 3-folds. In doing this I develop a new approach to study maximal singularities with positive dimensional centers. This allows to determine the pliability of a Q-factorial quartic with ordinary double points, and it shows the importance of Q-factoriality in the context of birational geometry of uniruled 3-folds, Comment: 16 pages, AMSLaTeX2e, this is the final version of a preprint that circulated in 2002 and benefited much from referee suggestions. To appear in Math. Ann

We prove that for smooth projective threefolds whose anticanonical divisors are nef, the second Chern classes are pseudo-effective under a weak assumption. As an application, the pseudo-effectivity of the second Chern classes implies that Kawamata's Effective Non-vanishing Conjecture holds for such threefolds., Comment: 18 pages

We study Calabi--Yau 3-folds with infinitely many divisorial contractions. We also suggest a method to describe Calabi--Yau 3-folds with the infinite automorphism group.

The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional components of this stratification are smooth rational curves (whose numerical equivalence classes are) called F-curves. The F-conjecture asserts that a divisor on $\M_{g,n}$ is ample if and only if it positively intersects the $F-$curves. In this paper the F-conjecture on $\M_{g,n}$ is reduced to showing that certain divisors in $\M_{0,N}$ for $N \leq g+n$ are equivalent to the sum of the canonical divisor plus an effective divisor supported on the boundary. As an application of the reduction, numerical criteria are given which if satisfied by a divisor $D$ on $\M_g$, show that $D$ is ample. Additionally, an algorithm is described to check that a given divisor is ample. Using a computer program called Nef Wizard, written by Daniel Krashen (http://www.yale.edu/users/dkrashen/nefwiz), one can use the criteria and the algorithm to verify the conjecture for low genus. This is done on $\M_g$ for $g \le 24$, more than doubling the known cases of the conjecture and showing it is true for the first genus such that $\M_g$ is known to be of general type., Comment: 20 pages

We extend the conjecture on the derived equivalence and K-equivalence to the logarithmic case and prove it in the toric case., Comment: 20 pages

We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb Q$-factorial toric varieties. So, our theory seems to be quite different from Reid's original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb Q$-factorial contractions, which imply that the $\mathbb Q$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Program., Comment: 21 pages; typos were corrected, new remarks were added

We investigate the structure of smooth projective 3-folds X with -K_X nef and K_X^3=0., Comment: 44 pages

Let $(X, \Delta)$ be a four-dimensional log variety that is projective over the field of complex numbers. Assume that $(X, \Delta)$ is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of "log quasi-numerically positive", by relaxing that of "numerically positive". Next we prove that, if the log canonical divisor $K_X + \Delta$ is log quasi-numerically positive on $(X, \Delta)$ then it is semi-ample., Comment: 4 pages, LaTeX2e, the published version