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Eric Jovinelly

Publications and source records attributed to Eric Jovinelly.

10 recordsLinked to original sources
arXiv2026

Finding large families of rational curves through Bend-and-Break

We present a new construction that allows us to break off large-degree rational curves from families of higher genus curves. Our construction and results deepen the connection between rational curves and positivity of the anticanonical divisor. Specifically, we show that varieties with large Fujita invariant admit large families of rational curves. We also construct free rational curves on certain singular Fano varieties. As an explicit consequence of our results, we prove that for a general Fano hypersurface of index at least 3, all spaces of genus g curves of sufficiently large degree have the expected dimension.

math.AG↗
arXiv2026

Stability of normal bundles of Brill--Noether curves in $\mathbb{P}^4$

We prove that a general Brill--Noether curve $C$ of genus $g \geq 2$ and degree $d$ in $\mathbb{P}^4$ has stable normal bundle $N_C$ if and only if $$(g, d) \notin \{(2,6), (3,7), (5,8), (6,9), (7,10)\}.$$ Moreover, $N_C$ is strictly semistable if $(g, d) \in \{(3, 7), (5, 8)\}$, and is unstable if $(g, d) \in \{(2, 6), (6, 9), (7, 10)\}$. Our results are valid in any characteristic. Along the way, we also generalize previous results of Larson--Vogt on interpolation for $N_C(-1)$, from characteristic zero to arbitrary characteristic.

math.AG↗
arXiv2026

Optimal bounds in Bend-and-Break

We improve the Bend-and-Break result of Miyaoka and Mori by establishing the optimal degree bound. Our result also yields optimal bounds on lengths of extremal rays of log canonical pairs.

math.AG↗
arXiv2026

Free curves and fundamental groups

We show that klt Fano varieties and certain lc Fano varieties contain free higher-genus curves in their smooth loci. Our methods also allow us to find free curves on varieties in positive characteristic and on quasiprojective varieties, under a natural positivity condition on the tangent bundle. We then use the existence of free curves to deduce finiteness of the fundamental group of the smooth locus in these settings. The paper includes an appendix by de Jong that establishes the Künneth formula for tame étale fundamental groups.

math.AG↗
arXiv2025

The integral chow ring of $M_2^{ct}$

This paper computes the integral Chow ring of the moduli space $M_2^{ct}$ of stable genus 2 curves of compact type. This is done by excising boundary strata from $\bar M_2$ one-by-one. During this process, we determine the Chow rings of all other open strata in $\bar M_2$ with $Z[1/2]$-coefficients.

math.AG↗
arXiv2024

Geometric Manin's Conjecture for Fano 3-Folds

We classify families of free rational curves on all smooth Fano threefolds over the complex numbers. In particular, we prove the family of very free rational curves representing any fixed numerical curve class is either irreducible or empty. This proves Geometric Manin's Conjecture in dimension three. For general Fano threefolds of each deformation type, our results allow us to explicitly count the number of components of the moduli space of irreducible, geometrically rational curves, which may not be free, representing any numerical class.

math.AG↗
arXiv2024

Rational Curves on Coindex 3 Fano Varieties

We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For varieties of dimension 4, we describe families of rational curves in terms of Fujita's $a$-invariant. Our results verify Lehmann and Tanimoto's Geometric Manin's Conjecture for all smooth coindex 3 Fano varieties over the complex numbers.

math.AG↗
arXiv2022

Rational Contractions of Fiber Type on $\overline{M}_{0,6}$

We identify a set of initial rational contractions of fiber type on $\overline{M}_{0,6}$. Our proof uses a new algorithm we develop for verifying descriptions of the cone of effective divisors on varieties without elementary rational contractions of fiber type. On Mori dream spaces, our algorithm identifies faces of the cone of nef curves associated to initial rational contractions of fiber type.

math.AG↗
arXiv2022

Extreme Divisors on $\bar{M}_{0,7}$ and Differences over Characteristic 2

We find 101,052 new extreme divisors on $\bar{M}_{0,7}$ (in 31 $S_7$-orbits) and millions of extreme nef curves over characteristic 0. Over characteristic 2, we identify two more $S_7$-orbits of extreme divisors, and prove $\bar{\text{Eff}}^k (\bar{M}_{0,n})$ is strictly larger over characteristic 2 than it is over characteristic 0, for all $1\leq k \leq n-6$. For each such $k$ we provide explicit cycles which are extreme in $\text{Eff}^k(\bar{M}_{0,n})$ over characteristic 2 but external to $\bar{\text{Eff}}^k(\bar{M}_{0,n})$ over characteristic 0. We apply our method of finding new extreme divisors to compute $\text{Eff}(\bar{M}_{0,\mathcal{A}})$ for $\mathcal{A}=(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, 1)$, proving it is polyhedral over any field, and conjecture a description of $\text{Eff}(\text{Bl}_e \bar{LM}_7)$.

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