A geometric characterisation of toric varieties

We prove a conjecture of Shokurov which characterises toric varieties using log pairs.Comment: 40 page

On the connectedness principle and dual complexes for generalized pairs

Let $(X,B)$ be a pair, and let $f \colon X \rightarrow S$ be a contraction with $-(K_X + B)$ nef over $S$. A conjecture, known as the Shokurov-Koll\'{a}r connectedness principle, predicts that $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two connected components, where $s \in S$ is an arbitrary schematic point and $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of $(X,B)$. In this work, we prove this conjecture, characterizing those cases in which $\mathrm{Nklt}(X,B)$ fails to be connected, and we extend these same results also to the category of generalized pairs. Finally, we apply these results and the techniques to the study of the dual complex for generalized log Calabi-Yau pairs, generalizing results of Koll\'{a}r-Xu and Nakamura.Comment: Minor correction

Rational curves and strictly nef divisors on Calabi--Yau threefolds

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve when its topological Euler characteristic is not $0$.Comment: 18 pages, comments are welcome

The Jordan property for local fundamental groups

We show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index one covers for $n$-dimensional klt singularities. We give an application to the study of local class groups of klt singularities.Comment: 24 page

A geometric characterization of toric varieties

We prove a conjecture of Shokurov which characterizes toric varieties using log pairs

Fano Varieties in Mori Fibre Spaces

We show that being a general fibre of a Mori fibre space is a rather restrictive condition for a Fano variety. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realised as a fibre of a Mori fibre space, which turn into a characterisation in the rigid case. We apply our criteria to figure out this property up to dimension three and on rational homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated

Hyperbolicity for log pairs

A classical result in birational geometry, Mori’s Cone Theorem, implies that if the canonical bundle of a variety X is not nef then X contains rational curves. This is the starting point of the so-called Minimal Model Program. In particular, hyperbolic varieties are positive from the point of view of birational geometry. Very much in the same vein, one could ask what happens for a quasi projective variety, Y . Using resolution of singularity, then one is lead to consider pairs (X,D) of a variety and a divisor, such that Y = X \\ D. I will show how to obtain a theorem analogous to Mori’s Cone Theorem in this context. Instead of rational complete curves, algebraic copies of the complex plane will male their appearance. I will also discuss an ampleness criterion for hyperbolic pairs.\r\nNon UBCUnreviewedAuthor affiliation: MITGraduat

Log geometry and extremal contractions

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, 2015.Cataloged from PDF version of thesis.Includes bibliographical references (pages 77-82).The Minimal Model Program (in short, MMP) aims at classifying projective algebraic varieties from a birational point of view. That means that starting from a projective algebraic variety X, [Delta] it is allowed to change the variety under scrutiny as long as its field of rational functions remains the same. In this thesis we study two problems that are inspired by the techniques developed in the last 30 years by various mathematicians in an attempt to realize the Minimal Model Program for varieties of any dimension. In the first part of the thesis, we prove a result about the existence and distribution of rational curves in projective algebraic varieties. We consider projective log canonical pairs (X,[Delta] A) where the locus Nklt(X,[Delta] A) of maximal singularities of the pair (X,[Delta] A) is nonempty. We show that if Kx[Delta]+ A is not nef then there exists an algebraic curves C, whose normalization is isomorphic to A1, contained either in X \ Nklt(X,[Delta] A) or in certain locally closed varieties that stratify Nklt(X,[Delta] A). This result implies a strengthening of the Cone Theorem for log canonical pairs. In the second part, we study certain varieties that naturally arise as possible outcomes of the classification algorithm proposed by the MMP. These are called Mori fibre spaces. A Mori fibre space is a variety X with log canonical singularities together with a morphism f : X --> Y such that the general fiber of f is a positive dimensional Fano variety and the monodromy of f is as large as possible. We show that being the general fiber of a Mori fiber space is a very restrictive condition for Fano varieties with terminal Q-factorial singularities. More specifically, we obtain two criteria (one sufficient and one necessary) for a Q-factorial Fano variety with terminal singularities to be realized as a fiber of a Mori fiber space. We apply our criteria to figure out what Fano varieties satisfy this property up to dimension three and to study the case of certain homogeneous spaces. The smooth toric case is studied and an interesting connection with K-semistability is also investigated.by Roberto Svaldi.Ph. D