National Science Foundation grants    Informal description of research interests

   Preprints, papers and books on Algebraic Geometry
  1. Categorical aspects of the Kollár-Shepherd-Barron correspondence, with Giancarlo, 43p. (2022), arXiv:2204.13225
    Abstract. We categorify the Milnor fiber of a smoothing of a 2-dimensional cyclic quotient singularity W to a hereditary algebra R, which can be viewed as a smoothing of the Kalck-Karmazyn algebra of W. The algebra R is semi-simple if and only if the smoothing is Q-Gorenstein (one direction is due to Kawamata). In general, a smoothing of W is dominated by a Q-Gorenstein smoothing of one of its M-resolutions (partial resolutions with Wahl singularities and nef canonical divisor). The algebra R is the endomorphism algebra of a strong exceptional collection of Hacking vector bundles associated to the N-resolution of W introduced in this paper. The N-resolution, which has a nef anticanonical divisor, is connected to the M-resolution via a geometric action of the braid group on all Wahl resolutions constructed using the universal family of k1A and k2A extremal neighborhoods. One of the consequences of the existence of this braid group action is that the categorified Milnor fiber is a hereditary algebra, and in particular is rigid in families. We give applications to the derived categories of Dolgachev surfaces.

  2. The BGMN conjecture via stable pairs, with Sebastian, 45p. (2021), arXiv:2108.11951
    Video introduction  Another video (by Sebastian)
    Abstract. Let C be a smooth projective curve of genus g>1 and let N be the moduli space of stable rank 2 vector bundles on C of odd degree. We construct a semi-orthogonal decomposition of the bounded derived category of N conjectured by Narasimhan and by Belmans, Galkin and Mukhopadhyay. It has two blocks for each i-th symmetric power of C for i=0,...,g-2 and one block for the (g-1)-st symmetric power. We conjecture that the subcategory generated by our blocks has a trivial semi-orthogonal complement, proving the full BGMN conjecture. Our proof is based on an analysis of wall-crossing between moduli spaces of stable pairs, combining classical vector bundles techniques with the method of windows.

  3. Blown-up toric surfaces with non-polyhedral effective cone, with Ana-Maria, Antonio and Luca U. 55 p. (2020), arXiv:2009.14298
    What is an effective cone?  Video introduction  A video by Ana-Maria 
    Abstract. We construct examples of projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone, both in characteristic 0 and in every prime characteristic p. As a consequence, we prove that the pseudo-effective cone of the Grothendieck--Knudsen moduli space of stable rational curves with n marked points is not polyhedral for n≥10 in characteristic 0 and in characteristic p, for all primes p. Many of these toric surfaces are related to a very interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Their analysis in characteristic p relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang-Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic p, for an infinite set of primes p of positive density.

  4. Scattering amplitudes of stable curves, 51 p. (2020), arXiv:2007.03831
    Seminar notes
    Abstract. Equations of hypertree divisors on the Grothendieck-Knudsen moduli space of stable rational curves, introduced by Castravet and Tevelev, appear as numerators of scattering amplitude forms for n massless particles in N=4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka. Rather than being a coincidence, this is just the tip of the iceberg of an exciting relation between algebraic geometry and high energy physics. We interpret scattering amplitude forms as probabilistic Brill-Noether theory: the study of statistics of images of n marked points under a random meromorphic function uniformly distributed with respect to the translation-invariant volume form of the Jacobian. We focus on the maximum helicity violating regime, which leads to a beautiful physics-inspired geometry for various classes of algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.

  5. Derived category of moduli of pointed curves - II, with Ana-Maria, 59 p. (2020), arXiv:2002.02889
    Abstract. We prove the Manin-Orlov conjecture: the moduli space of stable rational curves with n marked points has a full exceptional collection equivariant under the action of the symmetric group Sn permuting the marked points. In particular, its K-group with integer coefficients is a permutation Sn-lattice.

  6. Compactifications of moduli of points and lines in the projective plane, with Luca S. IMRN (2021), 79p.
    Introductory video Seminar notes
    Abstract. Projective duality identifies moduli spaces of points in P2 and lines in the dual P2. The latter space admits Kapranov's Chow quotient compactification, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of reducible degenerations of P2 with "broken lines". Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing reducible degenerations of P2 with n smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003.

  7. Spherical Tropicalization with Tassos, Transformation Groups 26 (2021), 691-718 (Ernest Vinberg memorial volume)
    Abstract. We extend tropicalization and tropical compactification of subvarieties of algebraic tori to subvarieties of spherical homogeneous spaces. Given a tropical compactification of a subvariety, we show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the subvariety. The proof is based on our equivariant version of the attening by blow-up theorem. We provide many examples.

  8. Exceptional collections on certain Hassett spaces, with Ana-Maria, Épijournal de Géométrie Algébrique 4 (2020), 1-34
    Abstract. We construct an S2xSn invariant full exceptional collection on Hassett spaces of weighted stable rational curves with n+2 markings and weights (1/2+a,1/2+a,b,...,b), for very small positive a,b, that can be identified with symmetric GIT quotients of (P1)n by the diagonal action of Gm when n is odd, and their Kirwan desingularization when n is even. The existence of such an exceptional collection is one of the needed ingredients in order to prove the existence of a full Sn-invariant exceptional collection on the moduli space of stable rational curves with n marked points . To prove exceptionality we use the method of windows in derived categories. To prove fullness we use previous work on the existence of invariant full exceptional collections on Losev-Manin spaces.

  9. Derived category of moduli of pointed curves - I, with Ana-Maria, Algebraic Geometry 7 (6) (2020), 722-757
    Abstract. This is the first paper in the sequence devoted to derived category of moduli spaces of curves of genus 0 with marked points. We develop several approaches to describe it equivariantly with respect to the action of the symmetric group permuting marked points. We construct an equivariant full exceptional collection on the Losev-Manin space which categorifies derangements.

  10. The Craighero-Gattazzo surface is simply-connected, with Julie and Giancarlo, Compositio, 153 (2017), 557-585
    Abstract. We show that the Craighero-Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply-connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero-Gattazzo surface is the only explicit example of a smooth simply-connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod p technique and deformation theory.

  11. Flipping Surfaces, with Paul and Giancarlo, Journal of Algebraic Geometry, 26 (2017), 279-345
    Abstract. We study semistable extremal threefold neighborhoods following earlier work of Mori, Kollar, and Prokhorov. We classify possible flips and extend Mori's algorithm for computing flips of extremal neighborhoods of type k2A to more general neighborhoods of type k1A. In fact we show that they belong to the same deformation family as k2A, and we explicitly construct the universal family of extremal neighborhoods. This construction follows very closely Mori's division algorithm, which we interpret as a sequence of mutations in the cluster algebra of rank 2 with general coefficients. We identify, in the versal deformation space of a cyclic quotient singularity, the locus of deformations such that the total space admits a (terminal) antiflip. We show that these deformations come from at most two irreducible components of the versal deformation space. As an application, we give an algorithm for computing stable one-parameter degenerations of smooth projective surfaces (under some conditions) and describe several components of the Kollar-Shepherd-Barron boundary of the moduli space of smooth canonically polarized surfaces of geometric genus zero.

  12. M 0,n is not a Mori Dream Space, with Ana-Maria, Duke Math. Journal, 164, no. 8 (2015), 1641-1667
    Abstract. Building on the work of Goto, Nishida and Watanabe on symbolic Rees algebras of monomial primes, we prove that the moduli space of stable rational curves with n punctures is not a Mori Dream Space for n>133. This answers the question of Hu and Keel.

  13. On a Question of Teissier, Collectanea Math., 65, no. 1 (2014), 61-66
    Abstract. We answer positively a question of B. Teissier on existence of resolution of singularities inside an equivariant map of toric varieties.

  14. Hypertrees, Projections, and Moduli of Stable Rational Curves , with Ana-Maria, Crelle's Journal, 675 (2013), 121-180.
    A database of hypertree divisors by Scheidwasser and Opie.
    Abstract. We give a description for the subcone of effective divisors of the Grothendieck-Knudsen moduli space of stable rational curves with n marked points. Namely, we introduce new combinatorial structures called hypertrees and show they give exceptional divisors with many remarkable properties.

  15. Rigid Curves on M 0,n and Arithmetic Breaks , with Ana-Maria, Contemporary Math., 564 (2012), 19-67
    Abstract. A result of Keel and McKernan states that a hypothetical counterexample to the F-conjecture must come from rigid curves on the moduli space of stable rational curves that intersect the interior. We exhibit several ways of constructing rigid curves. In all our examples, a reduction mod p argument shows that the classes of the rigid curves that we construct can be decomposed as sums of F-curves.

  16. Stable Pair, Tropical, and Log Canonical Compact Moduli of Del Pezzo Surfaces, with Paul and Sean, Inventiones Math. 178 , no.1 (2009), 173-228
    Abstract. We give a functorial normal crossing compactification of the moduli of smooth marked cubic surfaces entirely analogous to the Grothendieck-Knudsen moduli space of stable rationale curves

  17. Equations for M 0,n, with Sean, International Journal of Math. 20, no.9 (2009), 1--26
    Abstract. We show that the log canonical bundleof the moduli space of stable rational curves is very ample, show the homogeneous coordinate ring is Koszul, and give a nice set of rank 4 quadratic generators for the homogeneous ideal: The embedding is equivariant for the symmetric group, and the image lies on many Segre embedded copies of P1xP2x...xPn-3, permuted by the symmetric group. The homogeneous ideal of the moduli space is the sum of the homogeneous ideals of these Segre embeddings.

  18. Exceptional Loci on M 0,n and Hypergraph Curves , with Ana-Maria, 39p (2008), arXiv:0809.1699
    This preprint was largely superseded by [21] and [22], see also [32].

  19. Elimination Theory for Tropical Varieties, with Bernd, Math. Research Letters 15, no.3 (2008), 543-562
    Abstract. Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.

  20. The Newton Polytope of the Implicit Equation, with Bernd and Josephine, Moscow Math. Journal 7, no.2 (2007), 327-346
    Abstract. We apply tropical geometry to study the image of a map defined by Laurent polynomials with generic coefficients. If this image is a hypersurface then our approach gives a construction of its Newton polytope.

  21. Compactifications of Subvarieties of Tori, American Journal of Math. 129, no. 4 (2007), 1087-1104
    Abstract. We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including the moduli space of stable rational curves (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.

  22. Compactification of the Moduli Space of Hyperplane Arrangements, with Paul and Sean,   Journal of Algebraic Geometry 15 (2006), 657-680
    Abstract. Consider the moduli space M0 of arrangements of n hyperplanes in general position in projective (r-1)-space. When r=2 the space has a compactification given by the moduli space of stable curves of genus 0 with n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs (S,B) consisting of a variety S (possibly reducible) and a divisor B=B1+...+Bn satisfying various additional assumptions. We identify the closure of M0 in the moduli space of stable pairs as Kapranov's Chow quotient compactification of M0, and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.

  23. Hilbert's 14-th Problem and Cox Rings, with Ana-Maria, Compositio 142 (2006), 1479-1498
    Abstract. Our main result is the description of generators of the total coordinate ring of the blow-up of Pn in any number of points that lie on a rational normal curve. As a corollary we show that the algebra of invariants of the action of a two-dimensional vector group introduced by Nagata is finitely generated by certain explicit determinants. We also prove the finite generation of the algebras of invariants of actions of vector groups related to T-shaped Dynkin diagrams introduced by Mukai.

  24. Geometry of Chow Quotients of Grassmannians, with Sean, Duke Math. Journal 134, no. 2 (2006), 259-311
    Abstract. We consider Kapranov's Chow quotient compactification of the moduli space of ordered n-tuples of hyperplanes in Pr-1 in linear general position. For r=2 this is canonically identified with the Grothendieck-Knudsen moduli space of stable rational curves which has among others the nice properties 1) Modular meaning: stable pointed rational curves 2) Canonical description of limits of one parameter degenerations 3) Natural Mori theoretic meaning: log canonical compactification. We prove (1-2) generalize naturally to all (r,n), but that (3), which we view as the deepest, fails except possibly in the cases (2,n),(3,6),(3,7),(3,8), where we conjecture it holds.

  25. Projective Duality and Homogeneous Spaces, Springer 2005
    Abstract. For several centuries, various reincarnations of projective duality have inspired research in algebraic and differential geometry, classical mechanics, invariant theory, combinatorics, etc. On the other hand, projective duality is simply a systematic way of recoveringthe projective variety from the set of its tangent hyperplanes. In this survey we have tried to collect together different aspects of projective duality and points of view on it. We hope, that the exposition is quite informal and requires only a standard knowledge of algebraic geometry and algebraic (or Lie) group theory. Some chapters are, however, more difficult and use the modern intersection theory and homology algebra. But even in these cases we have tried to give simple examples and avoid technical difficulties.

Older papers on Transformation Groups and Invariant Theory