An example of Berglund-H\"ubsch mirror symmetry for a Calabi-Yau complete intersection

We study an example of complete intersection Calabi-Yau threefold due to Libgober and Teitelbaum arXiv:alg-geom/9301001, and verify mirror symmetry at a cohomological level. Direct computations allow us to propose an analogue to the Berglund-H\"ubsch mirror symmetry setup for this example arXiv:hep-th/9201014. We then follow the approach of Krawitz to propose an explicit mirror map arXiv:0906.0796.Comment: 18 pages, 4 table

On the boundedness of nn-folds with κ(X)=n−1\kappa(X)=n-1

In this note we study certain sufficient conditions for a set of minimal klt pairs $(X,\Delta)$ with $\kappa(X,\Delta)=\dim(X)-1$ to be bounded.Comment: Minor adjustment in the introductio

Rational points on 3-folds with nef anti-canonical class over finite fields

We prove that a geometrically integral smooth 3-fold $X$ with nef anti-canonical class and negative Kodaira dimension over a finite field $\mathbb{F}_q$ of characteristic $p>5$ and cardinality $q=p^e > 19$ 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 $b_1(X) \neq 0$ 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.Comment: 27 pages, comments are welcom

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

Moduli of Q\mathbb{Q}-Gorenstein pairs and applications

We develop a framework to construct moduli spaces of $\mathbb{Q}$-Gorenstein pairs. To do so, we fix certain invariants; these choices are encoded in the notion of $\mathbb{Q}$-stable pair. We show that these choices give a proper moduli space with projective coarse moduli space and they prevent some pathologies of the moduli space of stable pairs when the coefficients are smaller than $\frac{1}{2}$. Lastly, we apply this machinery to provide an alternative proof of the projectivity of the moduli space of stable pairs.Comment: Improved exposition and minor corrections throughout. Final version to appear in Journal of Algebraic Geometr