30 research outputs found
Rational points on 3‐folds with nef anti‐canonical class over finite fields
No full textWe prove that a geometrically integral smooth 3-fold X with nef anti-canonical class and negative Kodaira dimension over a finite field Fq of characteristic p>5 and cardinality q=pe>19 has a rational point. Additionally, under the same assumptions on p and q, we show that a 3-fold X with trivial canonical class and non-zero first Betti number b1(X)≠0 has a rational point. Our techniques rely on the Minimal Model Program to establish several structure results for generalized log Calabi--Yau 3-fold pairs over perfect fields
On the connectedness principle and dual complexes for generalized pairs
No full textLet be a pair, and let be a contraction
with nef over . A conjecture, known as the Shokurov-Koll\'{a}r
connectedness principle, predicts that has
at most two connected components, where is an arbitrary schematic
point and denotes the non-klt locus of . In this
work, we prove this conjecture, characterizing those cases in which
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: Final version, to appear in "Forum of Mathematics, Sigma
Boundedness of elliptic Calabi-Yau threefolds
No full textWe show that elliptic Calabi--Yau threefolds form a bounded family. We also
show that the same result holds for minimal terminal threefolds of Kodaira
dimension 2, upon fixing the rate of growth of pluricanonical forms and the
degree of a multisection of the Iitaka fibration. Both of these hypotheses are
necessary to prove the boundedness of such a family.Comment: Final version, to appear in J. Eur. Math. Soc. (JEMS
Arithmetic and geometric deformations of 3-folds
No full textWe show that mixed-characteristic and equi-characteristic small deformations
of 3-dimensional canonical (resp. terminal) singularities with perfect residue
field of characteristic are canonical (resp. terminal). We discuss
applications to arithmetic and geometric families of 3-dimensional Fano
varieties and minimal models with canonical singularities. Our results are
contingent upon the existence of log resolutions of 4-folds.Comment: v3: 19 pages, minor corrections. To appear in Bull. London Math. So
The Jordan property for local fundamental groups
No full textWe show the Jordan property for regional fundamental groups of klt singularities of fixed dimension. Furthermore, we prove the existence of effective simultaneous index 1 covers for n-dimensional klt singularities. We give an application to the study of local class groups of klt singularities
Generic Vanishing Fails for Surfaces in Positive Characteristic
No full textWe show that there exist smooth surfaces violating Generic Vanishing in any
characteristic . As a corollary, we recover a result of Hacon and
Kov\'acs, producing counterexamples to Generic Vanishing in dimension 3 and
higher.Comment: 12 pages, Bollettino dell'Unione Matematica Italiana (2017
Complements and coregularity of Fano varieties
No full textWe study the relation between the coregularity, the index of log Calabi-Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi-Yau pair of coregularity is at most, where is the Weil index of. This extends a recent result due to Filipazzi, Mauri and Moraga. We prove that a Fano variety of absolute coregularity admits either a -complement or a -complement. In the case of Fano varieties of absolute coregularity, we show that they admit an N-complement with N at most 6. Applying the previous results, we prove that a klt singularity of absolute coregularity admits either a -complement or -complement. Furthermore, a klt singularity of absolute coregularity admits an N-complement with N at most 6. This extends the classic classification of -type klt surface singularities to arbitrary dimensions. Similar results are proved in the case of coregularity. In the course of the proof, we prove a novel canonical bundle formula for pairs with bounded relative coregularity. In the case of coregularity at least, we establish analogous statements under the assumption of the index conjecture and the boundedness of B-representations.CA
Doctor of Philosophy
No full textdissertationThe focus of this dissertation is on birational geometry in characteristic zero. In particular, we consider the notion of generalized pairs, first introduced by Birkar and Zhang. As generalized pairs appear as the base of log Calabi-Yau fibrations, it is important to develop their theory and study their properties. This dissertation consists of two main parts, and each one of them investigates the properties of generalized pairs in a different direction. In the first part, which is the content of Chapter 5, we study some boundedness properties of generalized pairs. More precisely, we try to extend recent results of Hacon, McKernan and Xu about varieties of log general type to generalized pairs. In particular, we show that this extension is successful in the case of surfaces. The second main theme, discussed in Chapter 6, is the development of inductive methods in the study of log Calabi-Yau fibrations. We introduce a canonical bundle formula for generalized pairs. This tool allows analyzing log Calabi-Yau fibrations by breaking them into fibrations of smaller relative dimension or reducing them to have some explicit geometric properties. As an application, we prove some cases of a conjecture due to Prokhorov and Shokurov
