arXiv subjects

Maksym Fedorchuk

Publications and source records attributed to Maksym Fedorchuk.

26 records · Page 2Linked to original sources

arXiv2011

Finite Hilbert stability of canonical curves, II. The even-genus case

This paper is a sequel to arXiv:1109.4986, where we proved that a general smooth curve of odd genus, canonically or bicanonically embedded, has semistable finite Hilbert points. Here, we prove that a generic canonically embedded curve of even genus has semistable finite Hilbert points. More precisely, we prove that a generic canonically embedded trigonal curve of even genus has semistable finite Hilbert points. Furthermore, we show that the analogous result fails for bielliptic curves. Namely, the Hilbert points of bielliptic curves are asymptotically semistable but become non-semistable below a definite threshold value depending on genus.

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arXiv2011

Alternate Compactifications of Moduli Spaces of Curves

We give an informal survey, emphasizing examples and open problems, of two interconnected research programs in moduli of curves: the systematic classification of modular compactifications of $M_{g,n}$, and the study of Mori chamber decompositions of $\M_{g,n}$.

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arXiv2011

Singularities with G_m-action and the log minimal model program for $\bar{M}_g$

We give a precise formulation of the modularity principle for the log canonical models of $\bar{M}_g$. Assuming the modularity principle holds, we develop and compare two methods for determining the critical alpha-values at which a singularity or complete curve with G_m-action arises in the modular interpretations of log canonical models of $\bar{M}_g$. The first method involves a new invariant of curve singularities with G_m-action, constructed via the characters of the induced G_m-action on spaces of pluricanonical forms. The second method involves intersection theory on the variety of stable limits of a singular curve. We compute the expected alpha-values for large classes of singular curves, including curves with ADE, toric, and monomial unibranch Gorenstein singularities, as well as for ribbons, and show that the two methods yield identical predictions. We use these results to give a conjectural outline of the log MMP for $\bar{M}_g$.

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arXiv2011

Cyclic covering morphisms on $\bar{M}_{0,n}$

We study cyclic covering morphisms from $\bar{M}_{0,n}$ to moduli spaces of unpointed stable curves of positive genus or compactified moduli spaces of principally polarized abelian varieties. Our main application is a construction of new semipositive vector bundles and nef divisors on $\bar{M}_{0,n}$, with a view toward the F-conjecture. In particular, we construct new extremal rays of the symmetric nef cone of $\bar{M}_{0,n}$. We also find an alternate description of all sl level 1 conformal blocks divisors on $\bar{M}_{0,n}$.

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arXiv2010

Moduli spaces of hyperelliptic curves with A and D singularities

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two parameters describing allowable singularities. By varying these parameters, we interpolate between the stacks of stable limits of A and D singularities and the quotients of the versal deformation spaces of A and D singularities by a natural $\mathbb{G}_m$ action.

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arXiv2010

Moduli spaces of weighted stable curves and log canonical models of $\bar{M}_{g,n}$

Using Kollár's semipositivity results, we produce a number of nef and ample tautological divisors on Hassett's spaces of weighted stable pointed curves. As an application, we prove that Hassett's spaces are log canonical models of Deligne-Mumford spaces of stable pointed curves.

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arXiv2008

Ample divisors on moduli spaces of weighted pointed rational curves, with applications to log MMP for $\bar{M}_{0,n}$

We introduce a new technique for proving positivity of certain divisor classes on $\bar{M}_{0,n}$ and its weighted variants. Our methods give an unconditional description of the spaces of symmetric weighted pointed rational curves as log canonical models of $\bar{M}_{0,n}$.

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arXiv2007

Linear sections of the Severi variety and moduli of curves

We study the Severi variety $V_{d,g}$ of plane curves of degree $d$ and geometric genus $g$. Corresponding to every such variety, there is a one-parameter family of genus $g$ stable curves whose numerical invariants we compute. Building on the work of Caporaso and Harris, we derive a recursive formula for the degrees of the Hodge bundle on the families in question. For $d$ large enough, these families induce moving curves in $\bar{M}_g$. We use this to derive lower bounds for the slopes of effective divisors on $\bar{M}_g$. Another application of our results is to various enumerative problems on $V_{d,g}$.

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