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Any ample Cartier divisor D on a projective variety X is strictly nef (i.e. D.C>0 for any effective curve C on X). In general, the converse statement does not hold. But this is conjectured to be true for anticanonical divisors. The present paper establishes this fact for normal complex projective threefolds with canonical singularities. This result extends several previously known special cases. The proof rests mainly on sophisticated techniques of three dimensional birational geometry developed in the last two decades.
We prove a structure theorem for projective varieties with nef anticanonical divisors.
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 $c_3(X) eq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $ u(D) eq 1$, then $D$ is am
In this paper, we prove the ampleness conjecture and Serranos conjecture for strictly nef divisors on K-trivial fourfolds. Specifically, we show that any strictly nef divisors on projective fourfolds with trivial canonical bundle and vanishing irregularity are ample.
We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic
It was proved by J. A. Chen and M. Chen that a terminal Fano $3$-fold $X$ satisfies $(-K_X)^3geq frac{1}{330}$. We show that a non-rational $mathbb{Q}$-factorial terminal Fano $3$-fold $X$ with $rho(X)=1$ and $(-K_X)^3=frac{1}{330}$ is a weighted hyp