TerraciniLoci -- package for computing Terracini loci

Description

This package implements the algorithms from Section 8 of the paper Geometry of first nonempty Terracini loci by F. Galuppi, P. Santarsiero, D. Torrance, and E. Turatti.

The Terracini locus of projective variety $X$ is a subvariety of the symmetric power $X^{(r)}$ containing the closure of all sets $\{p_1,\ldots,p_r\}$ of smooth points in $X$ for which the space $\langle T_{p_1}X,\ldots,T_{p_r}X\rangle$ has less than the expected dimension.

This package exports one method, terraciniLocus, for computing the ideals of these varieties.

i1 : R = QQ[s,t]

o1 = R

o1 : PolynomialRing

i2 : S = QQ[x_0..x_3]

o2 = S

o2 : PolynomialRing

i3 : f = map(R, S, {s^3, s^2*t, s*t^2, t^3})

                  3   2      2   3
o3 = map (R, S, {s , s t, s*t , t })

o3 : RingMap R <-- S

i4 : terraciniLocus(2, f)

o4 = ideal 1

o4 : Ideal of QQ[z   ..z   ]
                  0,0   1,1

Authors

Version

This documentation describes version 0.6 of TerraciniLoci, released June 5, 2026.

Citation

If you have used this package in your research, please cite it as follows:

@article {MR5026390,
    AUTHOR = {Galuppi, Francesco and Santarsiero, Pierpaola and Torrance,
              Douglas A. and Turatti, Ettore Teixeira},
     TITLE = {Geometry of first nonempty {T}erracini loci},
   JOURNAL = {Commun. Contemp. Math.},
  FJOURNAL = {Communications in Contemporary Mathematics},
    VOLUME = {28},
      YEAR = {2026},
    NUMBER = {4},
     PAGES = {Paper No. 2550053},
      ISSN = {0219-1997,1793-6683},
   MRCLASS = {14J45 (14Q15 15A69)},
  MRNUMBER = {5026390},
       DOI = {10.1142/S0219199725500531},
       URL = {https://doi.org/10.1142/S0219199725500531},
}

Exports

Functions and commands
- terraciniLocus -- compute the Terracini locus of a projective variety
Methods
- terraciniLocus(ZZ,Ideal) -- see terraciniLocus -- compute the Terracini locus of a projective variety
- terraciniLocus(ZZ,Matrix,Ideal) -- see terraciniLocus -- compute the Terracini locus of a projective variety
- terraciniLocus(ZZ,RingMap) -- see terraciniLocus -- compute the Terracini locus of a projective variety

For the programmer

The object TerraciniLoci is a package, defined in TerraciniLoci.m2.

The source of this document is in /build/reproducible-path/macaulay2-1.26.06+ds/M2/Macaulay2/packages/TerraciniLoci.m2:174:0.