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Register a new userSets of Special Subvarieties of Bounded Degree
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David Urbanik
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Let $f : X to S$ be a family of smooth projective algebraic varieties over a smooth connected quasi-projective base $S$, and let $mathbb{V} = R^{2k} f_{*} mathbb{Z}(k)$ be the integral variation of Hodge structure coming from degree $2k$ cohomology it induces. Associated to $mathbb{V}$ one has the so-called Hodge locus $textrm{HL}(S) subset S$, which is a countable union of special algebraic subvarieties of $S$ parametrizing those fibres of $mathbb{V}$ possessing extra Hodge tensors (and so conjecturally, those fibres of $f$ possessing extra algebraic cycles). The special subvarieties belong to a larger class of so-called weakly special subvarieties, which are subvarieties of $S$ maximal for their algebraic monodromy groups. For each positive integer $d$, we give an algorithm to compute the set of all weakly special subvarieties $Z subset S$ of degree at most $d$ (with the degree taken relative to a choice of projective compactification $S subset overline{S}$ and very ample line bundle $mathcal{L}$ on $overline{S}$). As a corollary of our algorithm we prove conjectures of Daw-Ren and Daw-Javanpeykar-Kuhne on the finiteness of sets of special and weakly special subvarieties of bounded degree.
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Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The classical inequality S(n) $le$ R n (3) -- 2, between the Schur number S(n) and the Ramsey number R n (3) = R(3,. .. , 3), is shown to remain valid in a wider context, involving the Schur degree of certain subsets of G. Recursive upper bounds are known for R n (3) but not for S(n) so far. We formulate a conjecture which, if true, would fill this gap. Indeed, our study of the Schur degree leads us to conjecture S(n) $le$ n(S(n -- 1) + 1) for all n $ge$ 2. If true, it would yield substantially better upper bounds on the Schur numbers, e.g. S(6) $le$ 966 conjecturally, whereas all is known so far is 536 $le$ S(6) $le$ 1836.
More than four decades ago, Eisenbud, Khimv{s}iav{s}vili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $fcolon mathbb{A}^nto mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $mathbb{A}^1$-enumerative geometry.
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We determine the structure of the ring of Siegel modular forms of degree 2 in characteristic 3.
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