Papers

Growing Mathlib: maintenance of a large scale mathematical library, joint with Anne Baanen, Matthew Robert Ballard, Johan Commelin, Bryan Gin-ge Chen, Michael Rothgang, in Intelligent Computer Mathematics. CICM 2025. Lecture Notes in Computer Science, vol 16136.

Contact in algebraic and tropical geometry, joint with Marco Pacini, ZAG Handbook of Modern Algebraic Geometry, 2025.

Factors of HOMFLY polynomials, joint with Douglas Blackwell, Ann. Univ. Ferrara Sez. VII Sci. Mat. 70 (2024), no. 3, 1155–1163.

Complex and tropical counts via positive characteristic, joint with Marco Pacini, Expo. Math. 40 (2022), no. 4, 1096–1115.

The short version (curves are appropriately general):

• A plane quartic has \(28\) bitangent lines, unless it is tropical or defined over a field of characteristic \(2\), in which case it has \(7\).

• A plane curve of degree \(d\) has \(3 d (d-2)\) inflection lines, unless it is tropical or defined over a field of characteristic \(3\), in which case it has \(d(d-2)\).

• A canonical curve of genus \(g\) has \(2^{g-1} (2^g-1)\) theta-hyperplanes, unless it is tropical or defined over a field of characteristic \(2\), in which case it has \(2^g-1\).

We give a framework for analyzing these differences, by reduction to positive characteristic. As a consequence, we obtain divisibility properties for the numbers above:

• clearly, \(4\) divides \(28\);
• of course, \(3\) divides \(3 d (d-2)\);
• unsurprisingly, \(2^{g-1}\) divides \(2^{g-1} (2^g-1)\).

On a question of Dolgachev, joint with Marco Pacini, Math. Res. Lett., vol. 27, no. 1, pp. 281–299, 2020.

A classical construction in invariant theory associates to each quartic \( Q \) in \( \mathbf{P}^2 \), a quartic \( H(Q) \) in the dual projective plane \( (\mathbf{P}^2)^\vee \). Roughly, the points of \( H(Q) \) correspond to lines intersecting \( Q \) at a configuration of \(4\) points on \( \mathbf{P}^1 \) with vanishing \( j \)-invariant. Fix a general quartic \( Q’ \) in \( (\mathbf{P}^2)^\vee \). Dolgachev asked how many quartics \( Q \) in \( \mathbf{P}^2 \) are there with \( H(Q)=Q’ \). The answer is \(15\). The argument sheds no light on a possible interpretation of the number \(15\). (E.g. is \( 15 = 3 \cdot 5 = 2^4-1 = \binom{6}{2} = \cdots \)?).

Reconstructing general plane quartics from their inflection lines, joint with Marco Pacini, Trans. Amer. Math. Soc., vol. 372, no. 1, pp. 705–732, 2019.

We prove that the general plane quartic over a field of characteristic coprime with 6 is uniquely determined by its configuration of inflection lines.

We use the classical invariants of binary quartic forms, a degeneration argument and an explicit reconstruction!

A few questions about curves on surfaces, joint with Ciro Ciliberto, Andreas Leopold Knutsen, John Lesieutre, Victor Lozovanu, Rick Miranda, Yusuf Mustopa, Rend. Circ. Mat. Palermo (2) 66 (2017), no. 2, 195–204.

Curves in characteristic 2 with non-trivial 2-torsion, joint with Wouter Castryck and Marco Streng, Adv. Math. Commun. 8 (2014), no. 4, 479–495.

We study combinatorially defined theta-characteristics for curves in characteristic two.

Descent via (3,3)-isogeny on Jacobians of genus 2 curves, joint with Nils Bruin and Victor Flynn, Acta Arith. 165 (2014), no. 3, 201–223.

The infinite random simplicial complex, joint with Andrew Brooke-Taylor.

We introduce the infinite random simplicial complex \(\Delta\), a simplicial complex on a countable set of vertices.

The infinite random simplicial complex \(\Delta\) is the simplicial complex obtained almost surely by the following procedure.

For each pair of vertices, draw an edge between them with probability \(\frac{1}{2}\).

Proceeding inductively, suppose that the \((k-1)\)-skeleton \(\Delta_{k-1}\) of \(\Delta\) has already been constructed.

Let \(v_0,\ldots,v_k\) be a \((k+1)\)-tuple of vertices of \(\Delta_{k-1}\) with the property that every proper subset of \({v_0,\ldots,v_k }\) forms a face of \(\Delta_{k-1}\).

We add the \(k\)-dimensional face \({v_0,\ldots,v_k }\) to the \(k\)-skeleton \(\Delta_k\) of \(\Delta\) with probability \(\frac{1}{2}\).

On the unirationality of del Pezzo surfaces of degree two, joint with Cecília Salgado and Tony Várilly-Alvarado, J. Lond. Math. Soc. (2) 90 (2014), no. 1, 121–139.

We analyze in detail Manin’s unirationality construction for del Pezzo surfaces of degree two with a point, extending his results and clarifying an oversight.

We also show that del Pezzo surfaces of degree two over a finite field are unirational with at most three possible exceptions:

\[ \begin{array}{lrcl} X_1 / \mathbf{F}_3: & -w^2 & = & (x^2 + y^2)^2 + y^3z - yz^3, \\
X_2 / \mathbf{F}_3: & -w^2 & = & x^4 + y^3z - yz^3, \\
X_3 / \mathbf{F}_9: & \alpha w^2 & = & x^4 + y^4 + z^4, {\textrm{where }} \alpha \in \mathbf{F}_9 {\textrm{ is a non-square}}. \end{array} \]

Plane quartics with at least 8 hyperinflection points, joint with Marco Pacini, Bull. Braz. Math. Soc. (N.S.) 45 (2014), no. 4, 819–836.

We show that quartics over the complex numbers with at least eight hyperinflection lines are determined by their inflection lines.

The result uses the classification of quartics with at least eight hyperinflection lines by Vermeulen and exploits the fact that all these quartics have dihedral groups in their automorphism groups.

Our methods apply also in positive characteristic.

Amusingly we find that the three plane quartics over \(\mathbf{F}_{13}\) with equations

\[ x^4 + y^4 + z^4 + 3(x^2 y^2 + x^2 z^2 + y^2 z^2) = 0 \qquad x^4 + 3y^4 + 9z^4 + 3(9x^2 y^2 + 3x^2 z^2 + y^2 z^2) = 0 \qquad x^4 + 9y^4 + 3 z^4 + 3(3x^2 y^2 + 9x^2 z^2 + y^2 z^2) = 0 \] share the same inflection lines!

The same property holds for the two quartics \[ x^4 -y^4 -z^4 = 2x^2 y^2 -4xyz^2 \qquad -(x^4 -y^4 +z^4) = 2x^2 y^2 -4xyz^2 \] and we show that among the Vermeulen’s examples, these are essentially the only ones with this property.

On Büchi’s K3 surface, joint with Michela Artebani and Antonio Laface, Math. Z. 278 (2014), no. 3–4, 1113–1131.

We show that the K3 surface arising from Büchi’s problem is the Kummer surface of the Jacobian \(J\) of the genus two curve branched over an arithmetic progression of length 5.

We then exploit that the abelian surface \(J\) is isogenous to a product of elliptic curves to conclude that the set of rational points on Büchi’s K3 surface is Zariski dense.

Finally we give a natural modular interpretation of Büchi’s K3 surface as an irreducible component of the moduli space of rank two stable vector bundles on the Büchi K3 surface itself, opening the way for a modular solution to Büchi’s problem.

Erratum: In Theorem 3.7, the field \(K\) should be assumed infinite.

Finite Weil restriction of curves, joint with Victor Flynn, Monatsh. Math. 176 (2015), no. 2, 197–218.

We start with the following question: given a number field \(L\) and a rational function \(F(x)\) with coefficients in \(L\), when are there infinitely many values \(\alpha\) in \(L\) such that \(F(\alpha)\) is a rational number?

We generalize this question to the question of when does a fiber product of curves contain an irreducible component of genus at most one and we settle completely the problem when the curves in the fiber product have genus one.

Computing Néron-Severi groups and cycle class groups, joint with Bjorn Poonen and Ronald van Luijk, Compos. Math., 151 (2015), no. 4, 713–734.

We show that Néron-Severi groups (and more generally cycle class groups) are theoretically computable assuming the Tate conjecture and computability of certain étale cohomology groups.

Recovering plane curves of low degree from their inflection lines and inflection points, joint with Marco Pacini, Israel Journal of Mathematics, Volume 195 (Number 1), pp. 283–316.

We start investigating to what extent plane curves can be reconstructed from the knwoledge of their inflection lines.

For plane cubics, we prove that the reconstruction is always possible.

For general plane quartics, we show that the inflection lines and one inflection point are enough to reconstruct the curve.

Nef and semiample divisors on rational surfaces, joint with Antonio Laface, in “Torsors, étale homotopy and applications to rational points”, LMS Lecture Notes, edited by A. Skorobogatov, pp. 429–446.

We study the Cox rings of rational surfaces for which the anti-canonical divisor is (essentially) effective, finding conditions under which the Cox ring is finitely generated.

We show that there is a curve of arithmetic genus one and a finitely generated subgroup of its Picard group that is finite if and only if every nef divisor on the surface is semi-ample.

Thus we are able reduce the question of finite generation of Cox rings to the question of whether the semigroup of effective curves on the surface is finitely generated.

We conclude with an example of a Mori dream rational surface with vanishing anticanonical Iitaka dimension.

The surface parametrizing cuboids, joint with Michael Stoll.

We compute the Picard group of the surface \(S\) of cuboids; the emphasis is to prove that a very explicit set of 140 curves on \(S\) generates the full Picard group, not just a finite index subgroup. The main tool is the use of a combination of the geometric automorphism group of \(S\), as well as the Galois action on the set of curves to reduce the statement to the primitivity of the canonical class on \(S\).

Counting Rational Points on Cubic Curves, joint with Roger Heath-Brown, Sci. China Math. 53 (2010), no. 9, 2259–2268. Dedicated to Professor Wang Yuan.

By a combination of the determinant method and \(n\)-descent on curves of genus one, we find bounds for the number of rational points on a smooth plane cubic curve that are uniform in the coefficients and depend explicitly on the rank of the Mordell-Weil group of the curve. The argument exploits an interplay between analytic techniques to bound the number of solutions to a cubic equation and a geometric interpretation of an evaluation map on an abelian surface leading to the required inequalities.

Two-coverings of Jacobians of curves of genus two, joint with Victor Flynn and Ronald van Luijk, Proc. Lond. Math. Soc. (3) 104 (2012), no. 2, 387–429.

We construct explicit models for the two-coverings of Jacobians of genus two curves and also for some of their twists. An important role is played by the analysis of the action of the two-torsion subgroup on the equations of a natural embedding of the Jacobian of a curve of genus two in projective space, as well as the induced action on the Kummer variety.

Big rational surfaces, joint with Tony Várilly-Alvarado and Mauricio Velasco, Math. Ann. 351 (2011), no. 1, 95–107.

We present a unified approach to proving that smooth projective rational surfaces with big anti-canonical divisor have finitely generated Cox ring. Almost every previously known example was treated by ad hoc arguments on a case-by-case basis and is covered by our general result. Very few known examples of rational surfaces for which this statement was previously known are not covered by our result. We also provide many new examples. An explicit presentation in terms of generators and relations of the Cox ring of a variety is the main step in the explicit construction of a universal torsor on the variety itself.

Cox rings of degree one del Pezzo surfaces, joint with Tony Várilly-Alvarado and Mauricio Velasco, Algebra Number Theory 3 (2009), no. 7, 729–761.

We settle a conjecture of Batyrev and Popov on the ideal of relations of the Cox ring of a del Pezzo surface in the last remaining case of del Pezzo surfaces of degree one. Using techniques from commutative algebra, we transform the problem to a geometric question about the vanishing of certain cohomology groups of line bundles on the surface. We then analyze geometrically the various line bundles to reduce further the problem to a combinatorial question on configurations of exceptional curves on the surface. We finally solve the combinatorial question by explicit geometric arguments.

Fano varieties in index one Fano complete intersections, Math. Z. 259 (2008), no. 1, 61–64.

A generalization of a result of Roya Beheshti and Jason Starr. The results in this paper imply that, for \(n>4\), a smooth hypersurface of degree \(n\) in \(n\)-dimensional projective space cannot be covered by Fano varieties (resp. toric varieties) of dimension between 2 and \(n-3\) (inclusive). (For comparison, Roya and Jason prove the same statement but only for Fano subvarieties of dimension 2, rather than the full range.) This is supporting evidence to the widely believed conjecture that there exist rationally connected varieties that are not unirational and that, more precisely, general hypersurfaces of degree \(n\) in \(\mathbf{P}^n\) should be examples, for \(n\) large enough.

Conical and Spherical Graphs, joint with Mario Marietti, European J. Combin. 33 (2012), no. 7, 1606–1618. Dedicated to Tony Machì.

We introduce the notions of spherical and conical graphs, establishing links with (independent) dominating sets, edge covers and the homotopy type of associated simplicial complexes. We also prove a formula to compute the Euler characteristic of a simplicial set.

A simple uniform approach to complexes arising from forests, joint with Mario Marietti, Electron. J. Combin. 15 (2008), no. 1, Research Paper 101, 18 pp.

A follow up to the previous paper. We generalize our reduction technique to cover a wider range of simplicial complexes. While the complexes that we could analyze in the previous paper were either contractible or homotopy equivalent to spheres, the complexes here can also be (disjoint unions of) wedges of spheres. This added flexibility and generality has as a drawback that there are several discrete invariants associated to a given complex and thus a direct combinatorial characterization of the homotopy type is more difficult.

Cores of simplicial complexes, joint with Mario Marietti, Discrete Comput. Geom. 40 (2008), no. 3, 444–468.

A combinatorial paper. We introduce a “reduction technique” to study the homotopy type of simplicial complexes, with the goal of answering a question of Ehrenborg and Hetyei on the dimension of certain complexes known to be homotopy equivalent to spheres. As main applications we give combinatorial interpretations of these dimensions in several widely studied classes of graphs associated to forests.

Gröbner bases, monomial group actions, and the Cox rings of Del Pezzo surfaces, joint with Mike Stillman and Mauricio Velasco, J. Algebra 316 (2007), no. 2, 777–801.

My first paper on Cox rings. Using commutative algebra arguments, we exploit the action of the automorphism group of the Picard lattice of a del Pezzo surface to reduce a question of Batyrev and Popov on the Cox rings of del Pezzo surfaces to a completely mechanical verification. The main application is that we are able to use the computer to verify the conjecture for del Pezzo surfaces of degree at least three, a range that is still unreachable by brute force computations and that was also not known at the time. Later generalizations and computer-free proofs were found (also by myself!).

The irreducibility of the spaces of rational curves on del Pezzo surfaces, J. Algebraic Geom. 18 (2009), no. 1, 37–61. PhD Thesis, supervisor Johan de Jong.

The main result is the proof that the spaces of rational curves on del Pezzo surfaces representing a fixed divisor class are either empty or irreducible, with one exception. The argument involves a detailed analysis of the possible singularities of (a partial resolution of) the moduli spaces of rational curves on surfaces and combines this with a degeneration argument to reduce all the rational curves in a fixed divisor class to a standard form. The relevant moduli spaces are sufficiently smooth that we can show that the various degenerations always take place within the smooth locus, proving the required irreducibility. (The longer, actual PhD thesis is available here.)