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Register a new userTotal $p$-differentials on schemes over $Z/p^2$
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For a scheme $X$ defined over the length $2$ $p$-typical Witt vectors $W_2(k)$ of a characteristic $p$ field, we introduce total $p$-differentials which interpolate between Frobenius-twisted differentials and Buiums $p$-differentials. They form a sheaf over the reduction $X_0$, and behave as if they were the sheaf of differentials of $X$ over a deeper base below $W_2(k)$. This allows us to construct the analogues of Gauss-Manin connections and Kodaira-Spencer classes as in the Katz-Oda formalism. We make connections to Frobenius lifts, Borger-Weilands biring formalism, and Deligne--Illusie classes.
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Let $X$ be a set of $K$-rational points in $P^1 times P^1$ over a field $K$ of characteristic zero, let $Y$ be a fat point scheme supported at $ X$, and let $R_Y$ be the bihomogeneus coordinate ring of $Y$. In this paper we investigate the module of Kaehler differentials $Omega^1_{R_Y/K}$. We describe this bigraded $R_Y$-module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support $X$ is a complete intersection or an almost complete intersection in $P^1 times P^1$. Moreover, we introduce a Kaehler different for $Y$ and use it to characterize reduced fat point schemes in $P^1 times P^1$ having the Cayley-Bacharach property.
Let $p$ be a prime. Let $(R,ideal{m})$ be a regular local ring of mixed characteristic $(0,p)$ and absolute index of ramification $e$. We provide general criteria of when each abelian scheme over $Spec Rsetminus{ideal{m}}$ extends to an abelian scheme over $Spec R$. We show that such extensions always exist if $ele p-1$, exist in most cases if $ple ele 2p-3$, and do not exist in general if $ege 2p-2$. The case $ele p-1$ implies the uniqueness of integral canonical models of Shimura varieties over a discrete valuation ring $O$ of mixed characteristic $(0,p)$ and index of ramification at most $p-1$. This leads to large classes of examples of Neron models over $O$. If $p>2$ and index $p-1$, the examples are new.
We compute the expectation of the number of linear spaces on a random complete intersection in $p$-adic projective space. Here random means that the coefficients of the polynomials defining the complete intersections are sampled uniformly form the $p$-adic integers. We show that as the prime $p$ tends to infinity the expected number of linear spaces on a random complete intersection tends to $1$. In the case of the number of lines on a random cubic in three-space and on the intersection of two random quadrics in four-space, we give an explicit formula for this expectation.
We construct a local model for Hilbert-Siegel moduli schemes with $Gamma_1(p)$-level bad reduction over $text{Spec }mathbb{Z}_{q}$, where $p$ is a prime unramified in the totally real field and $q$ is the residue cardinality over $p$. Our main tool is a variant over the small Zariski site of the ring-equivariant Lie complex $_Aunderline{ell}_G^{vee}$ defined by Illusie in his thesis, where $A$ is a commutative ring and $G$ is a scheme of $A$-modules. We use it to calculate the $mathbb{F}_{q}$-equivariant Lie complex of a Raynaud group scheme, then relate the integral model and the local model.
We establish an uncertainty principle for functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ with constant support (where $p mid q-1$). In particular, we show that for any constant $S > 0$, functions $f: mathbb{Z}/p rightarrow mathbb{F}_q$ for which $|text{supp}; {f}| = S$ must satisfy $|text{supp}; hat{f}| = (1 - o(1))p$. The proof relies on an application of Szemeredis theorem; the celebrated improvements by Gowers translate into slightly stronger statements permitting conclusions for functions possessing slowly growing support as a function of $p$.
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