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Register a new userOn the strong maximal rank conjecture in genus 22 and 23
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We develop new methods to study tropicalizations of linear series and show linear independence of sections. Using these methods, we prove two new cases of the strong maximal rank conjecture for linear series of degree 25 and 26 on curves of genus 22 and 23, respectively.
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We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m=3, extending previous results for m=2.
We prove that the moduli spaces of curves of genus 22 and 23 are of general type. To do this, we calculate certain virtual divisor classes of small slope associated to linear series of rank 6 with quadric relations. We then develop new tropical methods for studying linear series and independence of quadrics and show that these virtual classes are represented by effective divisors.
Fix integers $r,s_1,...,s_l$ such that $1leq lleq r-1$ and $s_lgeq r-l+1$, and let $Cal C(r;s_1,...,s_l)$ be the set of all integral, projective and nondegenerate curves $C$ of degree $s_1$ in the projective space $bold P^r$, such that, for all $i=2,...,l$, $C$ does not lie on any integral, projective and nondegenerate variety of dimension $i$ and degree $<s_i$. We say that a curve $C$ satisfies the {it{flag condition}} $(r;s_1,...,s_l)$ if $C$ belongs to $Cal C(r;s_1,...,s_l)$. Define $ G(r;s_1,...,s_l)=maxleft{p_a(C): Cin Cal C(r;s_1,...,s_l)right }, $ where $p_a(C)$ denotes the arithmetic genus of $C$. In the present paper, under the hypothesis $s_1>>...>>s_l$, we prove that a curve $C$ satisfying the flag condition $(r;s_1,...,s_l)$ and of maximal arithmetic genus $p_a(C)=G(r;s_1,...,s_l)$ must lie on a unique flag such as $C=V_{s_1}^{1}subset V_{s_2}^{2}subset ... subset V_{s_l}^{l}subset {bold P^r}$, where, for any $i=1,...,l$, $V_{s_i}^i$ denotes an integral projective subvariety of ${bold P^r}$ of degree $s_i$ and dimension $i$, such that its general linear curve section satisfies the flag condition $(r-i+1;s_i,...,s_l)$ and has maximal arithmetic genus $G(r-i+1;s_i,...,s_l)$. This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.
We prove that a Shimura curve in the Siegel modular variety is not generically contained in the open Torelli locus as long as the rank of unitary part in its canonical Higgs bundle satisfies a numerical upper bound. As an application we show that the Coleman-Oort conjecture holds for Shimura curves associated to partial corestriction upon a suitable choice of parameters, which generalizes a construction due to Mumford.
We make a first geometric study of three varieties in $mathbb{C}^m otimes mathbb{C}^m otimes mathbb{C}^m$ (for each $m$), including the Zariski closure of the set of tight tensors, the tensors with continuous regular symmetry. Our motivation is to develop a geometric framework for Strassens Asymptotic Rank Conjecture that the asymptotic rank of any tight tensor is minimal. In particular, we determine the dimension of the set of tight tensors. We prove that this dimension equals the dimension of the set of oblique tensors, a less restrictive class introduced by Strassen.
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