An official website of the United States government
Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K .
more »
« less
Restricted Tangent Bundles for General Free Rational Curves
Abstract Suppose that $$X$$ is a smooth projective variety and that $$C$$ is a general member of a family of free rational curves on $$X$$. We prove several statements showing that the Harder–Narasimhan filtration of $$T_{X}|_{C}$$ is approximately the same as the restriction of the Harder–Narasimhan filtration of $$T_{X}$$ with respect to the class of $$C$$. When $$X$$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre’s “freeness” formulation of Manin’s Conjecture in the setting of rational curves.
more »
« less
- Award ID(s):
- 1945144
- PAR ID:
- 10367622
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- Volume:
- 2023
- Issue:
- 12
- ISSN:
- 1073-7928
- Page Range / eLocation ID:
- p. 9901-9949
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We prove that the moduli space of rational curves with cyclic action, constructed in our previous work, is realizable as a wonderful compactification of the complement of a hyperplane arrangement in a product of projective spaces. By proving a general result on such wonderful compactifications, we conclude that this moduli space is Chow-equivalent to an explicit toric variety (whose fan can be understood as a tropical version of the moduli space), from which a computation of its Chow ring follows.more » « less
-
We initiate the study of a class of real plane algebraic curves which we callexpressive. These are the curves whose defining polynomial has the smallest number of critical points allowed by the topology of the set of real points of a curve. This concept can be viewed as a global version of the notion of a real morsification of an isolated plane curve singularity. We prove that a plane curve is expressive if (a) each irreducible component of can be parametrized by real polynomials (either ordinary or trigonometric), (b) all singular points of in the affine plane are ordinary hyperbolic nodes, and (c) the set of real points of in the affine plane is connected. Conversely, an expressive curve with real irreducible components must satisfy conditions (a)–(c), unless it exhibits some exotic behaviour at infinity. We describe several constructions that produce expressive curves, and discuss a large number of examples, including: arrangements of lines, parabolas, and circles; Chebyshev and Lissajous curves; hypotrochoids and epitrochoids; and much more.more » « less
-
Abstract In 1977, Erd̋s and Hajnal made the conjecture that, for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ has a clique or stable set of size at least $$|G|^{c}$$, and they proved that this is true with $$ |G|^{c}$$ replaced by $$2^{c\sqrt{\log |G|}}$$. Until now, there has been no improvement on this result (for general $$H$$). We prove a strengthening: that for every graph $$H$$, there exists $c>0$ such that every $$H$$-free graph $$G$$ with $$|G|\ge 2$$ has a clique or stable set of size at least $$ \begin{align*} &2^{c\sqrt{\log |G|\log\log|G|}}.\end{align*} $$ Indeed, we prove the corresponding strengthening of a theorem of Fox and Sudakov, which in turn was a common strengthening of theorems of Rödl, Nikiforov, and the theorem of Erd̋s and Hajnal mentioned above.more » « less
-
Abstract Let $$\alpha \colon X \to Y$$ be a general degree $$r$$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $$r$$. We prove that the Tschirnhausen bundle of $$\alpha $$ is semistable if $$g(Y) \geq 1$$ and stable if $$g(Y) \geq 2$$.more » « less
