arXiv subjects

Eduardo Nascimento

Publications and source records attributed to Eduardo Nascimento.

3 recordsLinked to original sources

arXiv2025

Improved Bounds on Ultra-Log Concavity of the Grothendieck Class of $\overline{\mathcal{M}_{0,n}}$

The class of the fine moduli space of stable $n$-pointed curves of genus zero, $\overline{\mathcal{M}_{0,n}}$, in the Grothendieck ring of varieties encodes its Poincaré polynomial. Aluffi-Chen-Marcolli conjecture that the Grothendieck class of $\overline{\mathcal{M}_{0,n}}$ is real-rooted (and hence ultra-log-concave), and they proved an asymptotic ultra-log-concavity result for these polynomials. We build upon their work, by providing effectively computable bounds for the error term in their asymptotic formula for $\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})$. As a consequence, we prove that in the range $l \le \frac{n}{10\log n}$, the ultra-log-concavity inequality \[\left(\frac{\mathrm{rk}\, H^{2(l-1)}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-1}}\right)^2 \ge \frac{\mathrm{rk}\, H^{2(l-2)}(\overline{\mathcal{M}_{0,n}})\mathrm{rk}\, H^{2l}(\overline{\mathcal{M}_{0,n}})}{\binom{n-3}{l-2}\binom{n-3}{l}} \] holds for $n$ sufficiently large.

math.AG↗

arXiv2024

Explicit formulas for the Grothendieck class of $\overline{\mathcal M}_{0,n}$

We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space of genus 0 stable curves with n marked points. This information is equivalent to the Poincaré polynomial; it implies explicit expressions for the Betti numbers of the moduli space in terms of Stirling numbers or, alternatively, Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Grothendieck class as shown in work of Yuri Manin from the 1990s. This differential equation is equivalent to S. Keel's recursion for the Betti numbers of these moduli spaces. Our proof reduces the solution to two combinatorial identities which follow from applications of Lagrange series. We also study generating functions for the individual Betti numbers. In previous work it had been shown that these functions are determined by a set of polynomials with positive rational coefficients, which are conjecturally log-concave. We verify this conjecture for many infinite families of these polynomials, corresponding to the generating functions for the $2k$-Betti numbers of the moduli spaces for all $k\le 100$. Further, studying these polynomials allows us to prove that the generating function for the Grothendieck class of the moduli spaces may be written as a series of rational functions in the Lefschetz motive and the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of the moduli space.

math.AG↗

arXiv2023

Productive Crop Field Detection: A New Dataset and Deep Learning Benchmark Results

In precision agriculture, detecting productive crop fields is an essential practice that allows the farmer to evaluate operating performance separately and compare different seed varieties, pesticides, and fertilizers. However, manually identifying productive fields is often a time-consuming and error-prone task. Previous studies explore different methods to detect crop fields using advanced machine learning algorithms, but they often lack good quality labeled data. In this context, we propose a high-quality dataset generated by machine operation combined with Sentinel-2 images tracked over time. As far as we know, it is the first one to overcome the lack of labeled samples by using this technique. In sequence, we apply a semi-supervised classification of unlabeled data and state-of-the-art supervised and self-supervised deep learning methods to detect productive crop fields automatically. Finally, the results demonstrate high accuracy in Positive Unlabeled learning, which perfectly fits the problem where we have high confidence in the positive samples. Best performances have been found in Triplet Loss Siamese given the existence of an accurate dataset and Contrastive Learning considering situations where we do not have a comprehensive labeled dataset available.

cs.CV↗