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Register a new userPotential density of projective varieties having an int-amplified endomorphism
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We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with int-amplified endomorphisms in dimension $leq 3$. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.
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Let $X$ be a quasi-projective variety and $fcolon Xto X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $alpha_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $bar{kappa}(X)$ is non-negative (resp. the etale fundamental group $pi_1^{text{et}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $operatorname{deg}(f)geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with larger first dynamical degree (resp. every dominant endomorphism of any semiabelian variety).
We generalise Flo{}ystads theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphism to the projective space whose image is either arithmetically Cohen-Macaulay (ACM), or linearly normal and not contained in a quadric. We give necessary and sufficient conditions on integers $a$, $b$, and $c$ for a monad of type [ 0to(L^vee)^atomathcal{O}_{X}^{,b}to L^cto0 ] to exist. We show that under certain conditions there exists a monad whose cohomology sheaf is simple. We furthermore characterise low-rank vector bundles that are the cohomology sheaf of some monad as above. Finally, we obtain an irreducible family of monads over the projective space and make a description on how the same method could be used on an ACM smooth projective variety $X$. We establish the existence of a coarse moduli space of low-rank vector bundles over an odd-dimensional $X$ and show that in one case this moduli space is irreducible.
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projective varieties can be classified by symbol systems, a class of algebraic objects modeled on the systems of fundamental forms at general points of projective varieties. We study relations between the algebraic properties of symbol systems and the geometric properties of Euler-symmetric projective varieties. We describe also the relation between Euler-symmetric projective varieties of dimension n and equivariant compactifications of the vector group G_a^n.
For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1geq 1$ and $D$ is a very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(lambda)$ is the $d-1$ dimensional volume of ${overline {mathcal P}}_D cap {z= lambda}$, where ${overline {mathcal P}}_Dsubset {mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $kgeq 1$, the function $HKd(X, kD)$ can be replaced by another compactly supported continuous function $varphi_{kD}$ which is `linear in $k$. This gives the formula for the associated coordinate ring $(R, {bf m})$: $$lim_{kto infty}frac{e_{HK}(R, {bf m}^k) - e_0(R, {bf m}^k)/d!}{k^{d-1}} = frac{e_0(R, {bf m})}{(d-1)!}int_0^inftyvarphi_D(lambda)dlambda, $$ where $varphi_D$ (see Proposition~1.2) is solely determined by the shape of the polytope $P_D$, associated to the toric pair $(X, D)$. Moreover $varphi_D$ is a multiplicative function for Segre products. This yields explicit computation of $varphi_D$ (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope $P_D$, one can explicitly compute the limit for two dimensional toric pairs and their Segre products. We further show that (Theorem~6.3) the renormailzed limit takes the minimum value if and only if the polytope $P_D$ tiles the space $M_{mathbb R} = {mathbb R}^{d-1}$ (with the lattice $M = {mathbb Z}^{d-1}$). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.
We show various properties of smooth projective D-affine varieties. In particular, any smooth projective D-affine variety is algebraically simply connected and its image under a fibration is D-affine. In characteristic zero such D-affine varieties are also uniruled. We also show that (apart from a few small characteristics) a smooth projective surface is D-affine if and only if it is isomorphic to either ${mathbb P}^2$ or ${mathbb P}^1times {mathbb P}^1$. In positive characteristic, a basic tool in the proof is a new generalization of Miyaokas generic semipositivity theorem.
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