Interpolation for Brill--Noether curves
In this paper we determine the number of general points through which a Brill--Noether curve of fixed degree and genus in any projective space can be passed.
math.AG↗
arXiv subjects
Eric Larson
Publications and source records attributed to Eric Larson.
An Approach for Combining Multimodal Fusion and Neural Architecture Search Applied to Knowledge Tracing
Knowledge Tracing is the process of tracking mastery level of different skills of students for a given learning domain. It is one of the key components for building adaptive learning systems and has been investigated for decades. In parallel with the success of deep neural networks in other fields, we have seen researchers take similar approaches in the learning science community. However, most existing deep learning based knowledge tracing models either: (1) only use the correct/incorrect response (ignoring useful information from other modalities) or (2) design their network architectures through domain expertise via trial and error. In this paper, we propose a sequential model based optimization approach that combines multimodal fusion and neural architecture search within one framework. The commonly used neural architecture search technique could be considered as a special case of our proposed approach when there is only one modality involved. We further propose to use a new metric called time-weighted Area Under the Curve (weighted AUC) to measure how a sequence model performs with time. We evaluate our methods on two public real datasets showing the discovered model is able to achieve superior performance. Unlike most existing works, we conduct McNemar's test on the model predictions and the results are statistically significant.
cs.LG↗
Global Brill--Noether Theory over the Hurwitz Space
Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on $\mathbb{P}^1$.
math.AG↗
SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
SciPy is an open source scientific computing library for the Python programming language. SciPy 1.0 was released in late 2017, about 16 years after the original version 0.1 release. SciPy has become a de facto standard for leveraging scientific algorithms in the Python programming language, with more than 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories, and millions of downloads per year. This includes usage of SciPy in almost half of all machine learning projects on GitHub, and usage by high profile projects including LIGO gravitational wave analysis and creation of the first-ever image of a black hole (M87). The library includes functionality spanning clustering, Fourier transforms, integration, interpolation, file I/O, linear algebra, image processing, orthogonal distance regression, minimization algorithms, signal processing, sparse matrix handling, computational geometry, and statistics. In this work, we provide an overview of the capabilities and development practices of the SciPy library and highlight some recent technical developments.
cs.MS↗
Interpolation for Curves in Projective Space with Bounded Error
Given n general points p_1, p_2,..., p_n \in P^r, it is natural to ask whether there is a curve of given degree d and genus g passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if \[n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor.\] In the case of curves with nonspecial hyperplane section, the above conjecture was recently shown to hold with exactly three exceptions. In this paper, we prove a "bounded-error analog" for special linear series on general curves; more precisely we show that existance of such a curve subject to the stronger inequality \[n \leq \left\lfloor \frac{(r + 1)d - (r - 3)(g - 1)}{r - 1}\right\rfloor - 3.\] Note that the -3 cannot be replaced with -2 without introducing exceptions (as a canonical curve in P^3 can only pass through 9 general points, while a naive dimension count predicts 12). We also use the same technique to prove that the twist of the normal bundle N_C(-1) satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of general points contained in the hyperplane section of a general curve is at least \[\min\left(d, \frac{(r - 1)^2 d - (r - 2)^2 g - (2r^2 - 5r + 12)}{(r - 2)^2}\right).\] As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.
math.AG↗
Constructing Reducible Brill--Noether Curves
It was recently determined exactly through how many general points a nondegenerate curve with nonspecial hyperplane section can pass. This gives rise to a method of constructing reducible curves $C_1 \cup_ΓC_2 \to \mathbb{P}^r$ with general nodes: We take a finite set $Γ\subset \mathbb{P}^r$ of general points, and find nondegenerate nonspecial curves $C_1$ and $C_2$ in $\mathbb{P}^r$ of specified degrees and genera which pass through $Γ$, and glue together along $Γ$. The goal of this paper is to show that, subject to certain mild assumptions, stable maps constructed in this manner lie in the closure of the locus of nondegenerate stable maps from curves of general moduli, i.e. are BN-curves. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.
math.AG↗
The Integral Chow Ring of $\bar{M}_2$
In this paper we compute the Chow ring of the moduli stack $\bar{M}_2$ of stable curves of genus 2 with integral coefficients.
math.AG↗
Interpolation for Brill-Noether curves in $\mathbb{P}^4$
In this paper, we compute the number of general points through which a general Brill-Noether curve in $\mathbb{P}^4$ passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane. As explained in arXiv:1809.05980, these results play an essential role in the first author's proof of the Maximal Rank Conjecture.
math.AG↗
The Maximal Rank Conjecture for Sections of Curves
Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \to H^0(O_{(C_1 \cup C_2 \cup \cdots \cup C_n) \cap H}(m)) is of maximal rank, except for some counterexamples when m = 2. As explained in arXiv:1809.05980, this result plays a key role in the author's proof of the Maximal Rank Conjecture.
math.AG↗
Constructing Reducible Brill--Noether Curves II
In this paper, we study maps from reducible curves $f : C \cup_ΓD \to \mathbb{P}^r$. We restrict our attention to two cases: first, when $f|_D$ factors through a hyperplane $H$ and $f|_C$ is transverse to $H$; and second, when $r = 3$. Degeneration to stable maps of this type have played a crucial role in works of Hartshorne, Ballico, and others, on special cases of the maximal rank conjecture. However, the general problem of studying when such stable maps with specified combinatorial types exist remains open. Here, we give criteria for such Brill--Noether curves of this first type to exist, of specified degree $d$ and genus $g$, such that $f|_C$ is of specified degree $d'$ and genus $g'$. We also give criteria, sharpening earlier results of the author, for the existence of Brill--Noether space curves of specified combinatorial types. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture.
math.AG↗
The Maximal Rank Conjecture
Let C be a general curve of genus g, embedded in P^r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C.
math.AG↗
The Generality of a Section of a Curve
Let f: C --> P^3 be a general curve of genus g, mapped to P^3 via a general linear series of degree d; and let Q be a general (and thus smooth) quadric. In this paper, we show that the points of intersection f(C) \cap Q give a general collection of 2d points on Q, except for exactly six exceptional cases. We also prove similar theorems for every other pair (r, n) for which, except for only finitely many pairs (d, g), the intersection of a general curve of genus g mapped to P^r via a general linear series of degree d, with a general hypersurface S of degree n, is a general collection of dn points on S. As explained in arXiv:1809.05980, these results play a key role in the author's proof of the Maximal Rank Conjecture
math.AG↗
Degenerations of Curves in Projective Space and the Maximal Rank Conjecture
In this note, we give an overview of a new technique for studying Brill--Noether curves in projective space via degeneration. In particular, we give a roadmap to the proof of the Maximal Rank Conjecture.
math.AG↗
Interpolation for normal bundles of general curves
Given n general points p_1, p_2,..., p_n in P^r, it is natural to ask when there exists a curve C \subset P^r, of degree d and genus g, passing through p_1, p_2,..., p_n. In this paper, we give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N_C of a general nonspecial curve of degree d and genus g in P^r (with d >= g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H^0(N_C(-D)) = 0 or H^1(N_C(-D)) = 0), with exactly three exceptions.
math.AG↗
Interpolation for Restricted Tangent Bundles of General Curves
Let (C, p_1, p_2, \ldots, p_n) be a general marked curve of genus g, and q_1, q_2, ..., q_n \in P^r be a general collection of points. We determine when there exists a nondegenerate degree d map f : C \to P^r so that f(p_i) = q_i for all i. This is a consequence of our main theorem, which states that the restricted tangent bundle f^* T_{P^r} of a general curve of genus g, equipped with a general degree d map f to P^r, satisfies the property of interpolation (i.e.\ that for a general effective divisor D of any degree on C, either H^0(f^* T_{P^r}(-D)) = 0 or H^1(f^* T_{P^r}(-D)) = 0). We also prove an analogous theorem for the twist f^* T_{P^r}(-1).
math.AG↗
Determinants of Subquotients of Galois Representations Associated to Abelian Varieties
Given an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell$, the $\ell^n$-torsion points of $A$ give rise to a representation $ρ_{A, \ell^n} : \gal(\bar{K} / K) \to \gl_{2g}(\zz/\ell^n\zz)$. In particular, we get a mod-$\ell$ representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\ff_\ell)$and an $\ell$-adic representation $ρ_{A, \ell} : \gal(\bar{K} / K) \to \gl_{2g}(\zz_\ell)$. In this paper, we describe the possible determinants of subrepresentations (or more generally, subquotients) of these two representation for $\ell$ a prime number, as $A$ varies over all $g$-dimensional abelian varieties. Note that it is certainly not the case that any mod-$\ell$ subquotient lifts to an $\ell$-adic one. Nevertheless, the list of possible mod-$\ell$ characters turns out to be remarkably similar to the list of possible $\ell$-adic characters.
math.NT↗
On the Surjectivity of Galois Representations Associated to Elliptic Curves over Number Fields
Given an elliptic curve $E$ over a number field $K$, the $\ell$-torsion points $E[\ell]$ of $E$ define a Galois representation $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$. A famous theorem of Serre states that as long as $E$ has no Complex Multiplication (CM), the map $\gal(\bar{K}/K) \to \gl_2(\ff_\ell)$ is surjective for all but finitely many $\ell$. We say that a prime number $\ell$ is exceptional (relative to the pair $(E,K)$) if this map is not surjective. Here we give a new bound on the largest exceptional prime, as well as on the product of all exceptional primes of $E$. We show in particular that conditionally on the Generalized Riemann Hypothesis (GRH), the largest exceptional prime of an elliptic curve $E$ without CM is no larger than a constant (depending on $K$) times $\log N_E$, where $N_E$ is the absolute value of the norm of the conductor. This answers affirmatively a question of Serre.
math.NT↗
Progress Towards Counting D_5 Quintic Fields
Let $N(5,D_5,X)$ be the number of quintic number fields whose Galois closure has Galois group $D_5$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(5,D_5,X) \sim C X^{1/2}$ for some constant $C$. The best known upper bound is $N(5,D_5,X)\ll X^{3/4 + ε}$, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is $\ll X^{2/3}$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for $A_4$ quartic fields in terms of a similar norm equation.
math.NT↗